  
  [1X3 [33X[0;0YThe organization of the data[133X[101X
  
  [33X[0;0YWe  include  some  brief  comments  on  the  organization  of the data. As a
  preliminary step, we recall the [22Xp[122X-group generation algorithm.[133X
  
  
  [1X3.1 [33X[0;0YThe [22Xp[122X[101X[1X-group generation algorithm[133X[101X
  
  [33X[0;0YThe  [22Xp[122X-group  generation  algorithm  was developed and implemented by Eamonn
  O'Brien,  and  we  refer the reader to [O'B90] for a detailed description of
  the algorithm.[133X
  
  [33X[0;0YFor  a  brief  overview,  let  [22XP[122X  be a [22Xp[122X-group. The algorithm uses the lower
  [22Xp[122X-central     series,     defined     recursively     by     [22Xλ_1(P)=P[122X    and
  [22Xλ_i+1(P)=[λ_i(P),P]λ_i(P)^p[122X for [22Xi≥ 1[122X. The [22Xp[122X-class of [22XP[122X is the length of this
  series.  Each  [22Xp[122X-group  [22XP[122X,  apart  from  the  elementary abelian ones, is an
  immediate  descendant  of  the  quotient [22XP/R[122X where [22XR[122X is the last non-trivial
  term of the lower [22Xp[122X-central series of [22XP[122X.[133X
  
  [33X[0;0YThus  all  the  groups  of order [22X3^8[122X, except the elementary abelian one, are
  immediate  descendants of groups of order [22X3^k[122X for some [22Xk[122X smaller than [22X8[122X. All
  of  the  immediate  descendants  of  a  [22Xp[122X-group [22XQ[122X are quotients of a certain
  extension  of [22XQ[122X; the isomorphism problem for these descendants is equivalent
  to  the problem of determining orbits of certain subgroups of this extension
  under  an  action  of  the  automorphism  group  of [22XQ[122X. Not all [22Xp[122X-groups have
  immediate  descendants; those that do are called capable, and those which do
  not are called terminal.[133X
  
  [33X[0;0YO'Brien  and  Vaughan-Lee's  classification  of  the  groups of order [22Xp^7[122X in
  [OV05] is based on a classification of the nilpotent Lie rings of order [22Xp^7[122X,
  and  the  groups  of  order  [22Xp^7[122X  are  obtained from the Lie rings using the
  Baker-Campbell-Hausdorff  formula.  O'Brien  and  Vaughan-Lee classified the
  nilpotent  Lie  rings  of  order [22Xp^7[122X using the nilpotent Lie ring generation
  algorithm, which is a direct analogue of the [22Xp[122X-group generation algorithm.[133X
  
  [33X[0;0YThus  the databases of nilpotent Lie rings of order [22Xp^7[122X and of the groups of
  order  [22X3^8[122X  are  organized  according  to  these  algorithms:  the immediate
  descendants  of  order [22Xp^7[122X of each nilpotent Lie ring of order less than [22Xp^7[122X
  are  grouped  together  in  the  database  of  nilpotent  Lie rings, and the
  immediate  descendants of order [22X3^8[122X of each group of order less than [22X3^8[122X are
  grouped together in the database of groups of order [22X3^8[122X.[133X
  
  
  [1X3.2 [33X[0;0YThe groups of order 6561[133X[101X
  
  [33X[0;0YThe  database  of  groups  of  order  [22X3^8[122X is organized according to rank and
  [22Xp[122X-class.  Here the rank is the rank of the Frattini quotient, i.e., the size
  of  a  minimal generating set, and the [22Xp[122X-class is as defined in the previous
  section. The following table gives the number of groups of order [22X3^8[122X of each
  rank  and  [22Xp[122X-class, with the [22X(i,j)[122X entry corresponding to rank [22Xi[122X and [22Xp[122X-class
  [22Xj[122X.[133X
  
  [4X[32X[104X
    [4X  | 1   2        3        4       5      6     7   8[104X
    [4X--|-------------------------------------------------[104X
    [4X1 | 0   0        0        0       0      0     0   1[104X
    [4X2 | 0   0        58       486     1343   330   9   0[104X
    [4X3 | 0   4        216747   40521   2163   24    0   0[104X
    [4X4 | 0   23361    494666   22343   51     0     0   0[104X
    [4X5 | 0   578478   14796    80      0      0     0   0[104X
    [4X6 | 0   566      39       0       0      0     0   0[104X
    [4X7 | 0   10       0        0       0      0     0   0[104X
    [4X8 | 1   0        0        0       0      0     0   0[104X
  [4X[32X[104X
  
  [33X[0;0YIn the list of all groups of order [22X3^8[122X, the first group is the cyclic group,
  then  the  2-generator  groups follow in order of increasing [22Xp[122X-class, and so
  on.  The above table can thus be used to find the range of numbers of groups
  with a given rank and [22Xp[122X-class.[133X
  
  [33X[0;0YAs  mentioned  above,  the  database  is  organized according to the [22Xp[122X-group
  generation  algorithm.  For  example, the 9 groups of rank 2, [22Xp[122X-class 7, and
  order  [22X3^8[122X  are  numbered from 2219--2227. The groups numbered 2219 and 2220
  are  descendants  of SmallGroup([22X3^7,384[122X), and the groups numbered 2221--2227
  are  descendants of SmallGroup([22X3^7,386[122X). Similarly, the 24 groups of rank 3,
  [22Xp[122X-class 6, and order [22X3^8[122X are numbered from 261663--261686. The first four of
  these  groups  are  descendants  of  SmallGroup([22X3^7,5841[122X),  the  next 17 are
  descendants  of  SmallGroup([22X3^7,5844[122X),  and the last four are descendants of
  SmallGroup([22X3^7,5849[122X).[133X
  
  
  [1X3.3 [33X[0;0YThe groups with order the seventh power of a prime[133X[101X
  
  [33X[0;0YThe  groups  of  order  [22Xp^7[122X  for  primes [22Xp>11[122X are obtained from the LiePRing
  database  of  nilpotent  Lie  rings  of  order  [22Xp^7[122X  using Willem de Graaf's
  implementation   of   the  Baker-Campbell-Hausdorff  formula.  The  LiePRing
  database  is  organized  according to the output from the nilpotent Lie ring
  generation algorithm. For any given [22Xp[122X, the first Lie ring in the database is
  the  cyclic  Lie  ring  of order [22Xp^7[122X. Next come the two-generator Lie rings,
  then the three-generator Lie rings, and so on, ending with the six-generator
  Lie  rings,  and then finally the elementary abelian Lie ring of rank 7. The
  first  four  of  the  two-generator  nilpotent  Lie  rings  of order [22Xp^7[122X are
  immediate descendants of the Lie ring[133X
  
  
  [24X[33X[0;6Y\langle a,b|pb,class 3\rangle[133X
  
  [124X
  
  [33X[0;0Yof order [22Xp^4[122X. The next [22Xp^2+8p+25[122X are immediate descendants of the Lie ring[133X
  
  
  [24X[33X[0;6Y\langle a,b|baa,bab,pba,class 3\rangle[133X
  
  [124X
  
  [33X[0;0Yof   order   [22Xp^5[122X,  and  the  next  [22Xp+6+(p^2+3p+10)gcd(p-1,3)[122X  are  immediate
  descendants of[133X
  
  
  [24X[33X[0;6Y\langle a,b|babb,pa,pb,class 4\rangle .[133X
  
  [124X
  
  [33X[0;0YThe  nine  rank  6  Lie  rings in the database are the rank 6, [22Xp[122X-class 2 Lie
  rings.  These  are  the  immediate descendants of the elementary abelian Lie
  ring of rank 6.[133X
  
  [33X[0;0YThere  is  a  complete  list of presentations for the nilpotent Lie rings of
  order  [22Xp^k[122X  for  [22Xk≤  7[122X, valid for all [22Xp>3[122X, in the document [11Xp567.pdf[111X supplied
  with  the  documentation  for  the  LiePRing  package. The presentations are
  grouped  as  described  above,  with  each group of presentations giving the
  immediate descendants of a Lie ring of smaller order.[133X
  
  [33X[0;0YIn  a  few  cases  the descendants of a parametrized family of Lie rings are
  grouped  together.  For  example,  there  is a family of [22Xp(p-1)[122X distinct Lie
  rings with presentations of the form[133X
  
  
  [24X[33X[0;6Y\langle  a,b,c|ca-baa,cb,pa-\lambda  baa-\mu bab,pb+\nu baa+\xi bab,pc,class
  3\rangle[133X
  
  [124X
  
  [33X[0;0Ywith  [22Xλ  ,μ  ,ν  ,ξ  ≠  0[122X.  Most  of  these  Lie  rings  are  terminal,  but
  [22Xfrac52p-frac92+frac12gcd(p-1,4)[122X  of  them are capable, and they have a total
  of  [22Xfrac12p^3+2p^2-5p+frac12+fracp2gcd(p-1,4)[122X  descendants  of order [22Xp^7[122X and
  [22Xp[122X-class 4. These descendants have presentations[133X
  
  
  [24X[33X[0;6Y\langle              a,b,c|ca-baa,cb,pa-baa-\mu             bab-ybaaa,pb+\nu
  baa+bab-zbaaa,pc-tbaaa,class 4\rangle ,[133X
  
  [124X
  
  
  [24X[33X[0;6Y\langle     a,b,c|ca-baa,cb,pa-baa-\mu    bab-ybaaa,pb+\nu    baa+\mu    \nu
  bab-zbaaa,pc-tbaaa,class 4\rangle[133X
  
  [124X
  
  [33X[0;0Yfor various choices of the parameters [22Xμ ,ν ,y,z,t[122X. For any given value of [22Xp[122X,
  the   [22Xfrac12p^3+2p^2-5p+frac12+fracp2gcd(p-1,4)[122X   distinct  Lie  rings  with
  presentations of this form are grouped together, with consecutive numbering.[133X
  
  [33X[0;0YThere is no easy way to determine the numbering of (say) the three-generator
  Lie  rings of [22Xp[122X-class 4, since the numbers depend on [22Xp[122X in a very complicated
  way, and generally there is no easy, efficient way of searching the database
  for  a group with particular properties. In view of the numbers of groups of
  order  [22Xp^7[122X  and the time needed to generate a complete list, this means that
  the  database  will  be  of limited use for most people. A user who wants to
  access a particular batch of descendants as described in [11Xp567.pdf[111X is advised
  to  use the LiePRing package directly, as this package also has an option to
  obtain  the corresponding groups via Willem de Graaf's implementation of the
  Baker-Campbell-Hausdorff formula. On the other hand, there are only[133X
  
  
  [24X[33X[0;6Y2p^{3}+13p^{2}+64p+145+(p^{2}+10p+56)\gcd(p-1,3)+(4p+28)\gcd(p-1,4)[133X
  
  [124X
  
  
  [24X[33X[0;6Y+\;(2p+12)\gcd(p-1,5)+\gcd(p-1,7)+4\gcd(p-1,8)+\gcd(p-1,9)[133X
  
  [124X
  
  [33X[0;0Ytwo-generator  groups  of order [22Xp^7[122X for [22Xp>5[122X, and a complete list of them can
  be  generated  quite quickly for moderate values of [22Xp[122X. The table below gives
  the  number  of  [22Xd[122X-generator groups of order [22Xp^7[122X, valid for all [22Xp>5[122X, and the
  user  can use the table to compute the range of numbers needed to access the
  [22Xd[122X-generator groups of order [22Xp^7[122X for any given [22Xp[122X.[133X
  
  [8Xrank 1[108X
        [33X[0;6Y[22X1[122X[133X
  
  [8Xrank 2[108X
        [33X[0;6Y[22X2p^3+13p^2+64p+145+(p^2+10p+56)gcd(p-1,3)     +(4p+28)gcd(p-1,4)[122X     [22X+
        (2p+12)gcd(p-1,5)+gcd(p-1,7)+4gcd(p-1,8)+gcd(p-1,9)[122X[133X
  
  [8Xrank 3[108X
        [33X[0;6Y[22X2p^5+9p^4+29p^3+99p^2+380p+1100+(3p^2+28p+189)gcd(p-1,3)[122X             [22X+
        (p^2+13p+84)gcd(p-1,4)+(p+17)gcd(p-1,5)+gcd(p-1,8)+3gcd(p-1,7)[122X[133X
  
  [8Xrank 4[108X
        [33X[0;6Y[22Xp^5+3p^4+13p^3+57p^2+248p+1044+(6p+46)gcd(p-1,3)[122X                     [22X+
        (2p+23)gcd(p-1,4)+2gcd(p-1,5)[122X[133X
  
  [8Xrank 5[108X
        [33X[0;6Y[22Xp^2+15p+155[122X[133X
  
  [8Xrank 6[108X
        [33X[0;6Y[22X9[122X[133X
  
  [8Xrank 7[108X
        [33X[0;6Y[22X1[122X[133X
  
  [33X[0;0YThe total number of groups of order [22Xp^7[122X for all [22Xp > 5[122X is given by[133X
  
  
  [24X[33X[0;6Y3p^{5}+12p^{4}+44p^{3}+170p^{2}+707p+2455[133X
  
  [124X
  
  
  [24X[33X[0;6Y+\;(4p^{2}+44p+291)\gcd(p-1,3)+(p^{2}+19p+135)\gcd(p-1,4)[133X
  
  [124X
  
  
  [24X[33X[0;6Y+\;(3p+31)\gcd(p-1,5)+4\gcd(p-1,7)+5\gcd(p-1,8)+\gcd(p-1,9).[133X
  
  [124X
  
