The isomorphism problem asks whether a ring isomorphism RG ∼= RH implies that G ∼= H. This classical problem appears for the first time in regard to integral group rings in G. Higman’s Ph.D. thesis. In 1947, T. M. Thrall formulated it in the following terms: Given a group G and a field F , determine all groups H such that FG ∼= FH. It was first considered by S. Perlis and G. Walker in 1950 when they settled that finite abelian groups are determined by their rational group algebras. In this paper, pursuing an answer for the isomorphism problem in Thrall

sense, we classify the groups H with rational group algebra isomorphic to QG where G is a finite nilpotent class 2 group with cyclic center. Since all groups considered are finite and the classification is up to isomorphism we, most of the time, replace the condition QG ∼= QH by QG = QH. With this set, we can state our main theorem.