Finite groups have more conjugacy classes

We prove that for every $ϵ>0$ there exists a $\delta >0$ so that every group of order $n\ge 3$ has at least $\delta \log_{2} n/{(\log_{2} \log_{2} n)}^{3+\epsilon}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order $n$ has more than ${\log }_{3}n$ conjugacy classes. We answer Bertram’s question in the affirmative for groups with a trivial solvable radical.