Groups of order 78125

Statistics at a glance
Since $$78125 = 5^7$$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

GAP implementation
gap> SmallGroupsInformation(78125);

There are 34297 groups of order 78125.

E.A. O'Brien and M.R. Vaughan-Lee determined presentations of the groups with order p^7. A preprint of their paper is     available at      http://www.math.auckland.ac.nz/%7Eobrien/research/p7/paper-p7.pdf

For p in { 3, 5, 7, 11 } explicit lists of groups of order p^7 have been produced and stored into the database.

Giving the power commutator presentations of any of these groups using a standard notation they might be reduced to 35 elements of the group or a 245 p-digit number.

Only 56 of these digits may be unlike 0 for any group and even these 56 digits are mostly like 0. Further on these digits are often quite likely for sequences of subsequent groups. Thus storage of groups was done by finding a so     called head group and a so called tail. Along the tail only the different digits compared to the head are relevant. Even the tails occur more or less often and this is used to improve storage too. Since p^7 is too big the data is     stored into some remaing holes of SMALL_GROUP_LIB at      Primes[ p + 10 ].

This size belongs to layer 11 of the SmallGroups library. IdSmallGroup is not available for this size.