I am trying to answer the following: Let $G$ be a finite non-abelian group. Show that $k(G) > |Z(G)| + 1$, where $k(G) $ is the number of conjugacy classes of $G$.
I can see that each element in $Z(G) $ lies in a conjugacy class on its own. Also $G$ must have an element not in its centre because it is non-abelian and this element must be in another conjugacy class (which must contain at least 2 elements).
This gives $|Z(G)| + 1$ conjugacy classes so far. I am not sure how to show that there must be at least 1 more though.