The answer is **NO** for $G=SL_n(p)$ with $p$ fixed and $n\to\infty$. The bound you give - $(\log|G|)^c$ - requires an action of $G$ on a set of a size that is polynomial in $n$. But all the actions of $G$ have order exponential in $n$ - see, for instance Section 5 of Kleidman and Liebeck's book. I have an e-copy of this book if you want me to email it...