For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$.
E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in the Cayley graph of $G$.

What is known about the minimal genus of a surface faithfully acted upon by the symmetric group $S_n$?

(Same question may be asked for other families of finite groups.)