A digraph − → PC(G) is said to be the directed power graph on the conjugacy classes of a group G, if its vertices are the non-trivial conjugacy classes of G, and there is an arc from vertex C to C ′ if and only if C � = C ′ and C C ′m for some positive integer m > 0. Moreover, the simple graph PC(G) is said to be the (undirected) power graph on the conjugacy classes of a group G if its vertices are the conjugacy classes of G and two distinct vertices C and C ′ are adjacent in PC(G) if one is a subset of a power of the other. In this paper, we find some connections between algebraic properties of some groups and properties of the associated graph.