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

## Statement

### For a prime number

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

### More general version

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## Related facts

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

## Proof

### Proof case $p = 2$

Further information: element structure of dihedral groups, linear representation theory of dihedral groups

In this case, we can take $G$ to be any of the three maximal class groups of order $2^{m+2}$ (see classification of finite 2-groups of maximal class). For instance, we could take the dihedral group of order $2^{m+2}$ and degree $2^{m+1}$.

As per the linear representation theory of dihedral groups, all the degrees of irreducible representations are either 1 or 2, whereas the largest conjugacy class size is $2^m$.

### Proof for odd $p$

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