# Groups of order 625

## Contents

See pages on algebraic structures of order 625| 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-fourth order

Since $625 = 5^4$ 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 15 See classification of groups of prime-fourth order for odd prime
Number of abelian groups 5 equals the number of unordered integer partitions of $4$, which is the exponent in $5^4$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of nilpotency class exactly two 6 See classification of groups of prime-fourth order for odd prime
Number of groups of nilpotency class exactly three 4 See classification of groups of prime-fourth order for odd prime

## GAP implementation

The order 625 is part of GAP's SmallGroup library. Hence, any group of order 625 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 625 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(625);

There are 15 groups of order 625.
They are sorted by their ranks.
1 is cyclic.
2 - 10 have rank 2.
11 - 14 have rank 3.
15 is elementary abelian.

For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.```