Element structure of groups of order 32

Full listing
Here is the conjugacy class structure for all the groups of order 32:

Grouping by conjugacy class sizes
Here now is a grouping by conjugacy class sizes. Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the number of conjugacy classes are congruent to 32 mod 3, and hence congruent to 2 mod 3.

Grouping by cumulative conjugacy class sizes (number of elements)
It is true for this order that the cumulative conjugacy class size statistics values divide the order of the group in all cases. In fact, this is true in general when the order is $$p^k$$, with $$p$$ a prime and $$0 \le k \le 5$$. There are, however, counterexamples for $$2^6$$.


 * There exist groups of prime-sixth order in which the cumulative conjugacy class size statistics values do not divide the order of the group
 * All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order

Correspondence between conjugacy class sizes and degrees of irreducible representations
See also linear representation theory of groups of order 32

For groups of order 32, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Facts illustrated by these listings

 * Nilpotency class and order need not determine conjugacy class size statistics for groups of prime-fifth order
 * Number of conjugacy classes need not determine conjugacy class size statistics for groups of prime-fifth order
 * Conjugacy class size statistics need not determine nilpotency class for groups of prime-fifth order

Pairs where one of the groups is abelian
There are eight pairs of groups that are 1-isomorphic with the property that one of them is abelian. Of these, some pairs share the abelian group part.

Here is a summary version:

Here are the details:

For the first six of the cohomology perspective can be more compactly expressed as follows:

Grouping by abelian member
Of the seven abelian groups of order 32, five of them have non-abelian groups 1-isomorphic to them. The two missing ones are the obvious ones: cyclic group:Z32, on account of the fact that finite group having the same order statistics as a cyclic group is cyclic, and elementary abelian group:E32, on account of the fact that exponent two implies abelian.

Groupings that do not have any abelian member
These are groupings by 1-isomorphism where there are two or more members.

Order statistics raw data
Note that because number of nth roots is a multiple of n, we see that the number of elements whose order is $$1$$ or $$2$$ is odd, while all the other numbers are even. The total number of $$n^{th}$$ roots is even for all $$n = 2^k, k \ge 1$$.

Here is the GAP code to generate these order statistics:

gap> F := List(AllSmallGroups(32),G -> List(Set(G),Order));; gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)), > Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)), > Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x->x=16)),Length(Filtered(L,x ->x=32))]);; gap> M := List([1..51], i ->[i,K[i]]);

Here is GAP's output:

[ [ 1, [ 1, 1, 2, 4, 8, 16 ] ], [ 2, [ 1, 7, 24, 0, 0, 0 ] ], [ 3, [ 1, 3, 12, 16, 0, 0 ] ], [ 4, [ 1, 3, 12, 16, 0, 0 ] ], [ 5, [ 1, 7, 8, 16, 0, 0 ] ], [ 6, [ 1, 11, 20, 0, 0, 0 ] ], [ 7, [ 1, 11, 4, 16, 0, 0 ] ], [ 8, [ 1, 3, 12, 16, 0, 0 ] ], [ 9, [ 1, 11, 12, 8, 0, 0 ] ],  [ 10, [ 1, 3, 20, 8, 0, 0 ] ], [ 11, [ 1, 7, 16, 8, 0, 0 ] ], [ 12, [ 1, 3, 12, 16, 0, 0 ] ], [ 13, [ 1, 3, 20, 8, 0, 0 ] ],  [ 14, [ 1, 3, 20, 8, 0, 0 ] ], [ 15, [ 1, 3, 4, 24, 0, 0 ] ], [ 16, [ 1, 3, 4, 8, 16, 0 ] ], [ 17, [ 1, 3, 4, 8, 16, 0 ] ],  [ 18, [ 1, 17, 2, 4, 8, 0 ] ], [ 19, [ 1, 9, 10, 4, 8, 0 ] ], [ 20, [ 1, 1, 18, 4, 8, 0 ] ], [ 21, [ 1, 7, 24, 0, 0, 0 ] ],  [ 22, [ 1, 15, 16, 0, 0, 0 ] ], [ 23, [ 1, 7, 24, 0, 0, 0 ] ], [ 24, [ 1, 7, 24, 0, 0, 0 ] ], [ 25, [ 1, 11, 20, 0, 0, 0 ] ],  [ 26, [ 1, 3, 28, 0, 0, 0 ] ], [ 27, [ 1, 19, 12, 0, 0, 0 ] ], [ 28, [ 1, 15, 16, 0, 0, 0 ] ], [ 29, [ 1, 7, 24, 0, 0, 0 ] ],  [ 30, [ 1, 11, 20, 0, 0, 0 ] ], [ 31, [ 1, 11, 20, 0, 0, 0 ] ], [ 32, [ 1, 3, 28, 0, 0, 0 ] ], [ 33, [ 1, 7, 24, 0, 0, 0 ] ],  [ 34, [ 1, 19, 12, 0, 0, 0 ] ], [ 35, [ 1, 3, 28, 0, 0, 0 ] ], [ 36, [ 1, 7, 8, 16, 0, 0 ] ], [ 37, [ 1, 7, 8, 16, 0, 0 ] ],  [ 38, [ 1, 7, 8, 16, 0, 0 ] ], [ 39, [ 1, 19, 4, 8, 0, 0 ] ], [ 40, [ 1, 11, 12, 8, 0, 0 ] ], [ 41, [ 1, 3, 20, 8, 0, 0 ] ],  [ 42, [ 1, 11, 12, 8, 0, 0 ] ], [ 43, [ 1, 15, 8, 8, 0, 0 ] ], [ 44, [ 1, 7, 16, 8, 0, 0 ] ], [ 45, [ 1, 15, 16, 0, 0, 0 ] ],  [ 46, [ 1, 23, 8, 0, 0, 0 ] ], [ 47, [ 1, 7, 24, 0, 0, 0 ] ], [ 48, [ 1, 15, 16, 0, 0, 0 ] ], [ 49, [ 1, 19, 12, 0, 0, 0 ] ],  [ 50, [ 1, 11, 20, 0, 0, 0 ] ], [ 51, [ 1, 31, 0, 0, 0, 0 ] ] ] Here now are the cumulative order statistics:

Here are the GAP commands to generate the cumulative order statistics:

gap> F := List(AllSmallGroups(32),G -> List(Set(G),Order));; gap> J := List(F,L->[Length(Filtered(L,x -> x <= 1)), > Length(Filtered(L,x -> x <= 2)),Length(Filtered(L,x -> x <= 4)), > Length(Filtered(L,x -> x <= 8)),Length(Filtered(L,x->x<=16)),Length(Filtered(L,x ->x<=32))]);; gap> N := List([1..51], i ->[i,J[i]]);

Here is GAP's output:

[ [ 1, [ 1, 2, 4, 8, 16, 32 ] ], [ 2, [ 1, 8, 32, 32, 32, 32 ] ], [ 3, [ 1, 4, 16, 32, 32, 32 ] ], [ 4, [ 1, 4, 16, 32, 32, 32 ] ], [ 5, [ 1, 8, 16, 32, 32, 32 ] ], [ 6, [ 1, 12, 32, 32, 32, 32 ] ], [ 7, [ 1, 12, 16, 32, 32, 32 ] ], [ 8, [ 1, 4, 16, 32, 32, 32 ] ],  [ 9, [ 1, 12, 24, 32, 32, 32 ] ], [ 10, [ 1, 4, 24, 32, 32, 32 ] ], [ 11, [ 1, 8, 24, 32, 32, 32 ] ], [ 12, [ 1, 4, 16, 32, 32, 32 ] ],  [ 13, [ 1, 4, 24, 32, 32, 32 ] ], [ 14, [ 1, 4, 24, 32, 32, 32 ] ], [ 15, [ 1, 4, 8, 32, 32, 32 ] ], [ 16, [ 1, 4, 8, 16, 32, 32 ] ],  [ 17, [ 1, 4, 8, 16, 32, 32 ] ], [ 18, [ 1, 18, 20, 24, 32, 32 ] ], [ 19, [ 1, 10, 20, 24, 32, 32 ] ], [ 20, [ 1, 2, 20, 24, 32, 32 ] ],  [ 21, [ 1, 8, 32, 32, 32, 32 ] ], [ 22, [ 1, 16, 32, 32, 32, 32 ] ], [ 23, [ 1, 8, 32, 32, 32, 32 ] ], [ 24, [ 1, 8, 32, 32, 32, 32 ] ],  [ 25, [ 1, 12, 32, 32, 32, 32 ] ], [ 26, [ 1, 4, 32, 32, 32, 32 ] ], [ 27, [ 1, 20, 32, 32, 32, 32 ] ], [ 28, [ 1, 16, 32, 32, 32, 32 ] ],  [ 29, [ 1, 8, 32, 32, 32, 32 ] ], [ 30, [ 1, 12, 32, 32, 32, 32 ] ], [ 31, [ 1, 12, 32, 32, 32, 32 ] ], [ 32, [ 1, 4, 32, 32, 32, 32 ] ],  [ 33, [ 1, 8, 32, 32, 32, 32 ] ], [ 34, [ 1, 20, 32, 32, 32, 32 ] ], [ 35, [ 1, 4, 32, 32, 32, 32 ] ], [ 36, [ 1, 8, 16, 32, 32, 32 ] ],  [ 37, [ 1, 8, 16, 32, 32, 32 ] ], [ 38, [ 1, 8, 16, 32, 32, 32 ] ], [ 39, [ 1, 20, 24, 32, 32, 32 ] ], [ 40, [ 1, 12, 24, 32, 32, 32 ] ],  [ 41, [ 1, 4, 24, 32, 32, 32 ] ], [ 42, [ 1, 12, 24, 32, 32, 32 ] ], [ 43, [ 1, 16, 24, 32, 32, 32 ] ], [ 44, [ 1, 8, 24, 32, 32, 32 ] ],  [ 45, [ 1, 16, 32, 32, 32, 32 ] ], [ 46, [ 1, 24, 32, 32, 32, 32 ] ], [ 47, [ 1, 8, 32, 32, 32, 32 ] ], [ 48, [ 1, 16, 32, 32, 32, 32 ] ],  [ 49, [ 1, 20, 32, 32, 32, 32 ] ], [ 50, [ 1, 12, 32, 32, 32, 32 ] ], [ 51, [ 1, 32, 32, 32, 32, 32 ] ] ]

Equivalence classes based on order statistics
Here, we discuss the equivalence classes of groups of order 32 up to being order statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also order statistics-equivalent not implies 1-isomorphic.

Here is the GAP code to sort all groups of order 32 by equivalence classes:

gap> F := List(AllSmallGroups(32),G -> List(Set(G),Order));; gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)), > Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)), > Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x -> x = 16)), Length(Filtered(L,x -> x = 32))]);; gap> M := List([1..51], i -> [K[i],i]);; gap> S := SortedList(M);

Here is GAP's output:

[ [ [ 1, 1, 2, 4, 8, 16 ], 1 ], [ [ 1, 1, 18, 4, 8, 0 ], 20 ], [ [ 1, 3, 4, 8, 16, 0 ], 16 ], [ [ 1, 3, 4, 8, 16, 0 ], 17 ], [ [ 1, 3, 4, 24, 0, 0 ], 15 ], [ [ 1, 3, 12, 16, 0, 0 ], 3 ], [ [ 1, 3, 12, 16, 0, 0 ], 4 ], [ [ 1, 3, 12, 16, 0, 0 ], 8 ],  [ [ 1, 3, 12, 16, 0, 0 ], 12 ], [ [ 1, 3, 20, 8, 0, 0 ], 10 ], [ [ 1, 3, 20, 8, 0, 0 ], 13 ], [ [ 1, 3, 20, 8, 0, 0 ], 14 ],  [ [ 1, 3, 20, 8, 0, 0 ], 41 ], [ [ 1, 3, 28, 0, 0, 0 ], 26 ], [ [ 1, 3, 28, 0, 0, 0 ], 32 ], [ [ 1, 3, 28, 0, 0, 0 ], 35 ],  [ [ 1, 7, 8, 16, 0, 0 ], 5 ], [ [ 1, 7, 8, 16, 0, 0 ], 36 ], [ [ 1, 7, 8, 16, 0, 0 ], 37 ], [ [ 1, 7, 8, 16, 0, 0 ], 38 ],  [ [ 1, 7, 16, 8, 0, 0 ], 11 ], [ [ 1, 7, 16, 8, 0, 0 ], 44 ], [ [ 1, 7, 24, 0, 0, 0 ], 2 ], [ [ 1, 7, 24, 0, 0, 0 ], 21 ],  [ [ 1, 7, 24, 0, 0, 0 ], 23 ], [ [ 1, 7, 24, 0, 0, 0 ], 24 ], [ [ 1, 7, 24, 0, 0, 0 ], 29 ], [ [ 1, 7, 24, 0, 0, 0 ], 33 ],  [ [ 1, 7, 24, 0, 0, 0 ], 47 ], [ [ 1, 9, 10, 4, 8, 0 ], 19 ], [ [ 1, 11, 4, 16, 0, 0 ], 7 ], [ [ 1, 11, 12, 8, 0, 0 ], 9 ],  [ [ 1, 11, 12, 8, 0, 0 ], 40 ], [ [ 1, 11, 12, 8, 0, 0 ], 42 ], [ [ 1, 11, 20, 0, 0, 0 ], 6 ], [ [ 1, 11, 20, 0, 0, 0 ], 25 ],  [ [ 1, 11, 20, 0, 0, 0 ], 30 ], [ [ 1, 11, 20, 0, 0, 0 ], 31 ], [ [ 1, 11, 20, 0, 0, 0 ], 50 ], [ [ 1, 15, 8, 8, 0, 0 ], 43 ],  [ [ 1, 15, 16, 0, 0, 0 ], 22 ], [ [ 1, 15, 16, 0, 0, 0 ], 28 ], [ [ 1, 15, 16, 0, 0, 0 ], 45 ], [ [ 1, 15, 16, 0, 0, 0 ], 48 ],  [ [ 1, 17, 2, 4, 8, 0 ], 18 ], [ [ 1, 19, 4, 8, 0, 0 ], 39 ], [ [ 1, 19, 12, 0, 0, 0 ], 27 ], [ [ 1, 19, 12, 0, 0, 0 ], 34 ],  [ [ 1, 19, 12, 0, 0, 0 ], 49 ], [ [ 1, 23, 8, 0, 0, 0 ], 46 ], [ [ 1, 31, 0, 0, 0, 0 ], 51 ] ]