On the number of conjugacy classes of a primitive permutation group with nonabelian socle

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)<n\mathrm{/}2$ and $k(G)=o(n)$ as $n\to \mathrm{\infty }$, or $G$ belongs to explicit families of examples.