The conjugacy class number k(G) - a different perspective

Let $G$ be a finite group. Let $k(G)$ denote the number of conjugacy classes of $G$ and let $m(G)$ denote the least positive integer $n$ such that the union of any $n$ distinct non-trivial conjugacy classes of $G$ together with the identity of $G$ is a subgroup of $G$. We prove that $m(G)=k(G)-1$ for all $m(G)\ge 2$.