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

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 $$p^m$$. 

Related facts

 * Maximum conjugacy class size does not give bound on maximum degree of irreducible representation
 * 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 case $$p = 2$$
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 $$2^m$$.