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

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 $$p^m$$ but the maximum conjugacy class size in $$G$$ is $$p$$. 

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
Take the extraspecial group of order $$p^{2m + 1}$$ (there are two such groups and either will do). The maximum degree of irreducible representation for this group is $$p^m$$, and the maximum conjugacy class size is $$p$$.