Element structure of groups of prime-fifth order

Particular cases
The primes $$p = 2$$ and $$p = 3$$ behave somewhat differently from the other primes.

Grouping by cumulative conjugacy class sizes (number of elements)
Note that it is true in this case that the number of elements in conjugacy classes of size dividing any number itself divides the order of the group (in particular, all these numbers are powers of $$p$$). However, this is not true for all groups and in fact an analogous statement fails for groups of prime-sixth order (see element structure of groups of prime-sixth order). For more, see:


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

Correspondence between degrees of irreducible representations and conjugacy class sizes
See also linear representation theory of groups of prime-fifth order.

For groups of order $$p^5$$, it is true that the list of conjugacy class sizes determines the degrees of irreducible representations. In the case $$p = 2$$, the converse also holds, i.e., the degrees of irreducible representations determine the conjugacy class sizes. However, for $$p \ge 3$$, there is one ambiguous case: the case of $$p^2$$ degree one and $$p^3 - 1$$ degree two representations corresponds to two possible lists of conjugacy class sizes: ($$p$$ of size one, $$p^3 - 1$$ of size $$p$$, $$p^2 - p$$ of size $$p^3$$), and ($$p^2$$ of size 1, $$p^3 - 1$$ of size $$p^2$$). For $$p = 2$$, there are no groups fitting the latter case.

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
 * Degrees of irreducible representations need not determine conjugacy class size statistics