We include some brief comments on the organization of the data. As a preliminary step, we recall the p-group generation algorithm.
The p-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.
For a brief overview, let P be a p-group. The algorithm uses the lower p-central series, defined recursively by λ_1(P)=P and λ_i+1(P)=[λ_i(P),P]λ_i(P)^p for i≥ 1. The p-class of P is the length of this series. Each p-group P, apart from the elementary abelian ones, is an immediate descendant of the quotient P/R where R is the last non-trivial term of the lower p-central series of P.
Thus all the groups of order 3^8, except the elementary abelian one, are immediate descendants of groups of order 3^k for some k smaller than 8. All of the immediate descendants of a p-group Q are quotients of a certain extension of Q; 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 Q. Not all p-groups have immediate descendants; those that do are called capable, and those which do not are called terminal.
O'Brien and Vaughan-Lee's classification of the groups of order p^7 in [OV05] is based on a classification of the nilpotent Lie rings of order p^7, and the groups of order p^7 are obtained from the Lie rings using the Baker-Campbell-Hausdorff formula. O'Brien and Vaughan-Lee classified the nilpotent Lie rings of order p^7 using the nilpotent Lie ring generation algorithm, which is a direct analogue of the p-group generation algorithm.
Thus the databases of nilpotent Lie rings of order p^7 and of the groups of order 3^8 are organized according to these algorithms: the immediate descendants of order p^7 of each nilpotent Lie ring of order less than p^7 are grouped together in the database of nilpotent Lie rings, and the immediate descendants of order 3^8 of each group of order less than 3^8 are grouped together in the database of groups of order 3^8.
The database of groups of order 3^8 is organized according to rank and p-class. Here the rank is the rank of the Frattini quotient, i.e., the size of a minimal generating set, and the p-class is as defined in the previous section. The following table gives the number of groups of order 3^8 of each rank and p-class, with the (i,j) entry corresponding to rank i and p-class j.
| 1 2 3 4 5 6 7 8 --|------------------------------------------------- 1 | 0 0 0 0 0 0 0 1 2 | 0 0 58 486 1343 330 9 0 3 | 0 4 216747 40521 2163 24 0 0 4 | 0 23361 494666 22343 51 0 0 0 5 | 0 578478 14796 80 0 0 0 0 6 | 0 566 39 0 0 0 0 0 7 | 0 10 0 0 0 0 0 0 8 | 1 0 0 0 0 0 0 0
In the list of all groups of order 3^8, the first group is the cyclic group, then the 2-generator groups follow in order of increasing p-class, and so on. The above table can thus be used to find the range of numbers of groups with a given rank and p-class.
As mentioned above, the database is organized according to the p-group generation algorithm. For example, the 9 groups of rank 2, p-class 7, and order 3^8 are numbered from 2219--2227. The groups numbered 2219 and 2220 are descendants of SmallGroup(3^7,384), and the groups numbered 2221--2227 are descendants of SmallGroup(3^7,386). Similarly, the 24 groups of rank 3, p-class 6, and order 3^8 are numbered from 261663--261686. The first four of these groups are descendants of SmallGroup(3^7,5841), the next 17 are descendants of SmallGroup(3^7,5844), and the last four are descendants of SmallGroup(3^7,5849).
The groups of order p^7 for primes p>11 are obtained from the LiePRing database of nilpotent Lie rings of order p^7 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 p, the first Lie ring in the database is the cyclic Lie ring of order p^7. 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 p^7 are immediate descendants of the Lie ring
\langle a,b|pb,class 3\rangle
of order p^4. The next p^2+8p+25 are immediate descendants of the Lie ring
\langle a,b|baa,bab,pba,class 3\rangle
of order p^5, and the next p+6+(p^2+3p+10)gcd(p-1,3) are immediate descendants of
\langle a,b|babb,pa,pb,class 4\rangle .
The nine rank 6 Lie rings in the database are the rank 6, p-class 2 Lie rings. These are the immediate descendants of the elementary abelian Lie ring of rank 6.
There is a complete list of presentations for the nilpotent Lie rings of order p^k for k≤ 7, valid for all p>3, in the document p567.pdf 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.
In a few cases the descendants of a parametrized family of Lie rings are grouped together. For example, there is a family of p(p-1) distinct Lie rings with presentations of the form
\langle a,b,c|ca-baa,cb,pa-\lambda baa-\mu bab,pb+\nu baa+\xi bab,pc,class 3\rangle
with λ ,μ ,ν ,ξ ≠ 0. Most of these Lie rings are terminal, but frac52p-frac92+frac12gcd(p-1,4) of them are capable, and they have a total of frac12p^3+2p^2-5p+frac12+fracp2gcd(p-1,4) descendants of order p^7 and p-class 4. These descendants have presentations
\langle a,b,c|ca-baa,cb,pa-baa-\mu bab-ybaaa,pb+\nu baa+bab-zbaaa,pc-tbaaa,class 4\rangle ,
\langle a,b,c|ca-baa,cb,pa-baa-\mu bab-ybaaa,pb+\nu baa+\mu \nu bab-zbaaa,pc-tbaaa,class 4\rangle
for various choices of the parameters μ ,ν ,y,z,t. For any given value of p, the frac12p^3+2p^2-5p+frac12+fracp2gcd(p-1,4) distinct Lie rings with presentations of this form are grouped together, with consecutive numbering.
There is no easy way to determine the numbering of (say) the three-generator Lie rings of p-class 4, since the numbers depend on p 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 p^7 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 p567.pdf 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
2p^{3}+13p^{2}+64p+145+(p^{2}+10p+56)\gcd(p-1,3)+(4p+28)\gcd(p-1,4)
+\;(2p+12)\gcd(p-1,5)+\gcd(p-1,7)+4\gcd(p-1,8)+\gcd(p-1,9)
two-generator groups of order p^7 for p>5, and a complete list of them can be generated quite quickly for moderate values of p. The table below gives the number of d-generator groups of order p^7, valid for all p>5, and the user can use the table to compute the range of numbers needed to access the d-generator groups of order p^7 for any given p.
1
2p^3+13p^2+64p+145+(p^2+10p+56)gcd(p-1,3) +(4p+28)gcd(p-1,4) + (2p+12)gcd(p-1,5)+gcd(p-1,7)+4gcd(p-1,8)+gcd(p-1,9)
2p^5+9p^4+29p^3+99p^2+380p+1100+(3p^2+28p+189)gcd(p-1,3) + (p^2+13p+84)gcd(p-1,4)+(p+17)gcd(p-1,5)+gcd(p-1,8)+3gcd(p-1,7)
p^5+3p^4+13p^3+57p^2+248p+1044+(6p+46)gcd(p-1,3) + (2p+23)gcd(p-1,4)+2gcd(p-1,5)
p^2+15p+155
9
1
The total number of groups of order p^7 for all p > 5 is given by
3p^{5}+12p^{4}+44p^{3}+170p^{2}+707p+2455
+\;(4p^{2}+44p+291)\gcd(p-1,3)+(p^{2}+19p+135)\gcd(p-1,4)
+\;(3p+31)\gcd(p-1,5)+4\gcd(p-1,7)+5\gcd(p-1,8)+\gcd(p-1,9).
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