Groups of order 625
This article gives information about, and links to more details on, groups of order 625
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
|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 , which is the exponent in . 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|
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.