- The p-group generation algorithm
- The groups of order 6561
- The groups with order the seventh power of a prime

We include some brief comments on the organisation of the data. As a
preliminary step we briefly 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 OBrien90 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 `lambda _{1}(P)=P` and

Thus all the groups with order `3 ^{8}`, except the elementary abelian one,
are immediate descendants of groups with order

O'Brien and Vaughan-Lee's classification of the groups of order `p ^{7}`
in OVL05 is based on a classification of the nilpotent Lie rings of
order

Thus the databases of nilpotent Lie rings of order
`p ^{7}` and of the groups of order

The database of groups of order `3 ^{8}` is organized according to rank and

| 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

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(

The groups of order `p ^{7}` for

`
< a,b|pb,class 3 >
`

of order `p ^{4}`. The next

`
< a,b|baa,bab,pba,class 3 >
`

of order `p ^{5}`, and the next

`
< a,b|babb,pa,pb,class 4 >.
`

And so on. 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

In a few cases the descendants of a parametrized family of Lie rings are grouped together.

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

`rank 1`-
`1` -
`rank 2` -
`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)` -
`rank 3` -
`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) `rank 4`-
`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)` `rank 5`-
`p`^{2}+15p+155 `rank 6`-
`9` `rank 7`-
`1`

The total number of groups of order `p ^{7}` for all

`
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)+4gcd (p-1,7)+5gcd (p-1,8)+gcd (p-1,9).
`

sglppow manual

Dezember 2014