Let p be a prime number and G be a finite nonabelian p-group. The group G is called a CGZ-group if and only if every nonabelian subgroup H of G satisfies C_G(H) = Z(H). In this paper, we study the isomorphism classes of CGZ-groups of exponent p. More precisely, we prove that any CGZ-group G of order pⁿ and exponent p (where 3 ≤ n ≤ p) admits a unique characteristic elementary abelian subgroup of index p. Furthermore, we study the isomorphism classes of splits extensions of an abelian group by a cyclic group. As an application, we compute the number of isomorphism classes of CGZ-groups of order pⁿ and exponent p.

CGZ-group, p-group of maximal class, isomorphism classes, elementary abelian subgroup.