Groups of order 32

Numbers of groups
Since $$32 = 2^5$$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Summary information
Here, the rows are arithmetic functions that take values between $$0$$ and $$5$$, and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal $$51$$. To view a list of all groups, click on the value in the cell and the list of all groups with GAP IDs appears.

Up to isoclinism
The information below collects groups based on the equivalence relation of being isoclinic groups. The equivalence classes are also called Hall-Senior families.

Up to isologism for class two
Under the equivalence relation of being isologic groups with respect to the variety of groups of nilpotency class two, the equivalence classes are as follows (the table is incomplete):

Up to isologism for class three
Under the equivalence relation of being isologic groups with respect to the variety of groups of class at most three, there are two equivalence classes:

Up to isologism for higher class
For class four or higher, all groups of order 32 are isologic to each other.

GAP implementation
gap> SmallGroupsInformation(32);

There are 51 groups of order 32. They are sorted by their ranks. 1 is cyclic. 2 - 20 have rank 2. 21 - 44 have rank 3. 45 - 50 have rank 4. 51 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.