For any group G, let C(G) denote the set of centralizers of G. We say that a group G is a Cn-group if |C(G)| = n. We prove that every arbitrary Cn-group is center-by-(finite of order ≤ max{(n − 2)2, 2(n − 3)log (n−3) 2 }). Also we show that the derived length of a nilpotent Cn-group is at most [n − 1/2] + 1.
