Certain groups with bounded movement having the maximal number of orbits

Let G be a permutation group on a set Ω such that G has no fixed points in Ω. If, for a given positive integer m, the cardinalities Γg Γ is at most m for all g G and Γ Ω, then G is said to have bounded movement m on Ω. When the maximum of Γg Γ over all g G and Γ Ω is equal to m, we say G has bounded movement equal to m. We will show that if G has bounded movement equal to m and p ( 5) is the least odd prime dividing G, then it has at most 2m−(p−1) nontrivial orbits. Moreover the groups G attaining the maximum bound will be classified.