# Maximum conjugacy class size does not give bound on maximum degree of irreducible representation

From Groupprops

## Statement

### For a prime number

Suppose is a prime number. Then, for any positive integer , it is possible to construct a finite p-group such that the maximum degree of irreducible representation for is but the maximum conjugacy class size in is .

### More general version

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## Related facts

- Maximum degree of irreducible representation does not give bound on maximum conjugacy class size
- Degrees of irreducible representations need not determine conjugacy class size statistics
- Conjugacy class size statistics need not determine degrees of irreducible representations

For more related facts, see the facts section of the degrees of irreducible representations page.

## Proof

`Further information: element structure of extraspecial groups, linear representation theory of extraspecial groups`

Take the extraspecial group of order (there are two such groups and either will do). The maximum degree of irreducible representation for this group is , and the maximum conjugacy class size is .