Conjugacy class size statistics need not determine nilpotency class for groups of prime-fifth order

Statement

Suppose $p$ is 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 conjugacy class size statistics but have different Nilpotency class (?) values.

Proof

Case $p=2$

Further information: Element structure of groups of order 32#Conjugacy class sizes

There are two Hall-Senior families (i.e., equivalence classes up to isoclinism) of groups of order 32, both of which have the same conjugacy class size statistics:

• The family $\Gamma _{3}$ comprises ten groups, all of which have nilpotency class exactly three. An example is wreath product of Z4 and Z2.
• The family $\Gamma _{4}$ comprises nine groups, all of which have nilpotency class exactly two. An example is generalized dihedral group for direct product of Z4 and Z4.

In both families, all groups have the following conjugacy class size statistics: 4 of order 1, 6 of order 2, 4 of order 4.