# Groups of order 3125

## Contents

This article gives information about, and links to more details on, groups of order 3125
See pages on algebraic structures of order 3125| See pages on groups of a particular order

## Statistics at a glance

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

Since $3125 = 5^5$ is a prime power, and prime power order implies nilpotent, all groups of order 3125 are nilpotent groups.

Quantity Value Explanation
Total number of groups up to isomorphism 77 PORC function for number of groups of order $p^5$ for $p \ge 5$ is: $2p + 61 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$. Plugging in $p = 5$ gives 77.
Number of abelian groups 7 equals the number of unordered integer partitions of $5$ (this is the $5$ appearing in the exponent part of $5^5$). See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of nilpotency class exactly two 30 the general formula for order $p^5$ with $p \ge 5$ is $p + 25$.
Number of groups of nilpotency class exactly three 31 the general formula for order $p^5$ with $p \ge 5$ is $p + 26$.
Number of groups of nilpotency class exactly four (i.e., maximal class groups) 9 the general formula for order $p^5$ with $p \ge 5$ is $3 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$.

## GAP implementation

The order 3125 is part of GAP's SmallGroup library. Hence, any group of order 3125 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 3125 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(3125);

There are 77 groups of order 3125.
They are sorted by their ranks.
1 is cyclic.
2 - 38 have rank 2.
39 - 70 have rank 3.
71 - 76 have rank 4.
77 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 4 of the SmallGroups library.
IdSmallGroup is available for this size.```