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

From Groupprops

## Statement

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

## Related facts

### 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

## Proof

### Case

`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, , , and , 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 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 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 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.