Small Conjugacy Class and Large Character Degrees

Question

Let $G$ be a finite group, $m$ its minimum non-trivial character degree and $s$ a conjugacy class size. If $m$ is large, can $s$ be small?

Answer 1

Yes, for example ${\rm SL}(2,p)$ for odd $p$ has minimal nontrivial character degree $(p-1)/2$ but has centre of order $2$, so it has a nontrivial conuygacy class of size $1$.

But if $s>1$ then $s>m$, which you can see by considering the permutation representation of $G$ on the elements of the conjugacy class.