Conjugacy classes — counting them, and what centrality really gives

Consider the following statements: (I) If G is a finite group that has only 2 conjugate classes, then $|G|=2$. (II) If G is a group and $a\in Z(G)$, the center of G, then the conjugacy class of $a$ is $\{e,a\}$, where $e$ is the identity of G.