Let $G$ be a finite group and $S=\{g_{1}, g_{2},...,g_{k}\}$ be a system of representatives for its conjugacy classes. For $1\leq i\leq k$, let $C_G(g_{i}):=\{x\in G | xg_{i}=g_{i}x \}$ be the centralizer of $g_{i}$.
How could we obtain from $S$ a system of representatives for the conjugacy classes of $C_G(g_{i})$?.
Thank you so much!