# Groups of order 15625

## Contents

This article gives information about, and links to more details on, groups of order 15625
See pages on algebraic structures of order 15625| See pages on groups of a particular order

## Statistics at a glance

To understand these in a broader context, see: groups of order 5^n|groups of prime-sixth order

Since $15625 = 5^6$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity Value Explanation
Total number of groups 684
Number of abelian groups 11 Equals the number of unordered integer partitions of $6$, which is the exponent in $5^6$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of nilpotency class exactly two 149
Number of groups of nilpotency class exactly three 386
Number of groups of nilpotency class exactly four 99
Number of groups of nilpotency class exactly five (i.e., maximal class groups) 39

## GAP implementation

The order 15625 is part of GAP's SmallGroup library. Hence, any group of order 15625 can be constructed using the SmallGroup function by specifying its group ID. Unfortunately, IdGroup is not available for this order, i.e., given a group of this order, it is not possible to directly query GAP to find its GAP ID.

Further, the collection of all groups of order 15625 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

```gap> SmallGroupsInformation(15625);

There are 684 groups of order 15625.

Easterfield (1940) constructed a list of the groups of
order p^6 for p >= 5.

The database of parametrised presentations for the groups
with order p^6 for p >= 5 is based on the Easterfield
list, corrected by Newman, O'Brien and Vaughan-Lee (2004).
It differs only in the addition of groups in isoclinism
family \$Phi_{13}\$, in using the James (1980) presentations
for the groups in \$Phi_{19}\$, and a small number of
typographical amendments. The linear ordering employed is
very close to that of Easterfield.

Each group with order \$p^6\$ is described by a power-
commutator presentation on 6 generators and 21 relations:
15 are commutator relations and 6 are power relations.
Each presentation has the prime \$p\$ as a parameter.
The database contains about 500 parametrised
presentations, most of these have \$p\$ as the only
parameter.

This size belongs to layer 9 of the SmallGroups library.
IdSmallGroup is not available for this size.```