Groups with few conjugacy classes

Let G be a finite group, let p be a prime divisor of the order of G and let k(G) be the number of conjugacy classes of G. By disregarding at most finitely many non-solvable p-solvable groups G, we have $k(G)\geq2\smash{\sqrt{p-1}}$ with equality if and only if if $\smash{\sqrt{p-1}}$ is an integer, $G=C_{p}\rtimes\smash{C_{\sqrt{p-1}}}$ and CG(Cp) = Cp. This extends earlier work of Héthelyi, Külshammer, Malle and Keller.