# Groups of order 125

## Contents

See pages on algebraic structures of order 125| 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-cube order

Since $125 = 5^3$ 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 5 See classification of groups of prime-cube order
Number of abelian groups 3 Equals the number of unordered integer partitions of $3$, which is the exponent term in $5^3$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of nilpotency class exactly two 2

## GAP implementation

The order 125 is part of GAP's SmallGroup library. Hence, any group of order 125 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 125 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(125);

There are 5 groups of order 125.
1 is of type c125.
2 is of type 5x25.
3 is of type 5^2:5.
4 is of type 25:5.
5 is of type 5^3.

The groups whose order factorises in at most 3 primes
have been classified by O. Hoelder. This classification is
used in the SmallGroups library.

This size belongs to layer 1 of the SmallGroups library.
IdSmallGroup is available for this size.```