A permutation group H acting on a set X is said to be graphical if there is a graph G such that Γ(G), the automorphism group of G, is identical to H. Characterisation of graphical permutation groups seems to be difficult. Kagno and Chao have shown that the group generated by a single m-cycle is not graphical. Here we study the group generated by a permutation such that it consists of disjoint cycles whose lengths are multiples of the length of one of its cycles. Our results are obtained by constructing certain graphs which we call generalised permutation graphs. We also study graphical cycle permutation groups of order pm where p is a prime.