If a nilpotent group G acts faithfully on a solvable group H, it turned out to be helpful to know the orbit sizes of H in this action. Suppose that a nilpotent group G acts faithfully and irreducibly on V. It is well known that Z(G) is cyclic and the intersection of CG(v) and Z(G) equals to 1 for any nontrivial element v in V. Let G be a nilpotent group of class 2 with Z(G) cyclic. If S is a subgroup of G with |S|2>|G|, then the intersection of S and Z(G) is not trival. If G acts faithfully and irreducibly on an elementary abelian N, then the minimal orbit has size large than |G|1/2.