Groups with few class sizes and the centraliser equality subgroup

LetG be ap-group whose conjugacy classes have at mostk sizes. We prove thatG is abelian-by-(exponentp ^k−1) (ifp=2, exponent 2 ^k−2). It follows that a 2-group with three class sizes is metabelian. Various other results on class sizes are proved, and some conjectures are formulated.