Nilpotency class and order need not determine conjugacy class size statistics for groups of prime-fifth order

Statement
Let $$p$$ be a prime number. It is possible to have two groups $$P_1$$ and $$P_2$$, both of order $$p^5$$, such that $$P_1$$ and $$P_2$$ have the same nilpotency class but have different conjugacy class size statistics.

Similar facts

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

Opposite facts

 * Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

Case $$p = 2$$
For groups of order 32, there are three different Hall-Senior families of groups, $$\Gamma_2$$, $$\Gamma_4$$, and $$\Gamma_5$$, all of which comprise groups of nilpotency class two, but with the groups in each family having different conjugacy class size statistics from each other:


 * The family $$\Gamma_2$$ contains 15 groups, such as direct product of D8 and V4, and all groups in this family have conjugacy class sizes as follows: 8 conjugacy classes of size 1, 12 conjugacy classes of size 2.
 * The family $$\Gamma_4$$ contains 9 groups, such as generalized dihedral group for direct product of Z4 and Z4, and all groups in this family have conjugacy class sizes as follows: 4 conjugacy classes of size 1, 6 conjugacy classes of size 2, 4 conjugacy classes of size 4.
 * The family $$\Gamma_5$$ contains 2 groups, namely, the two extraspecial groups of order 32 (inner holomorph of D8 and central product of D8 and Q8) and both have conjugacy class sizes as follows: 2 conjugacy classes of size 1, 15 conjugacy classes of size 2.