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

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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.

Related facts

Similar facts

Opposite facts

Proof

Case p = 2

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

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.