A group G is said to be n-central if the factor group G/Z(G) is of exponent n. We improve a result of Gupta and Rhemtulla by showing that every 4-central group is 16-abelian and every 6-central group is 36-abelian. There are examples of finite groups which show that these bounds are best possible. Consequently, we can completely describe the structure of exponent semigroups of free non-cyclic n-central groups for n = 2, 3, 4, 6. We obtain a characterization of metabelian p-central groups and a classification of finitely generated 2-central groups. We compute the nilpotency class of the free metabelian 4-central group of arbitrary finite rank.