Let $G$ be a simple group, $|G|=n, p$ a prime such that $p|n$. If $G$ has more than $n/(p^2)$ conjugacy classes then the $p$-Sylow sub are abelian

Question

Let $G$ be a simple group, $|G|=n$, $p$ a prime such that $p\mid n$. Prove that if $G$ has more than $n/(p^2)$ conjugacy classes then the $p$-Sylow subgroups are abelian.

I think I have to use Sylow's theorem and irreducible characters but I'm not able to do It. I know the number of conjugacy classes is the same as that of irreducible characters and I think this can be the key to solving the problem, someone has an idea?