Let be a group and ∈ (). An α-commutator of elements , ∈ is defined as [ , ] = . In 2015, Barzegar et al. introduced an α-commutator of elements of and defined a new generalization of nilpotent groups by using the definition of α- commutators which is called an α-nilpotent group. They also introduced an α-commutator subgroup of , denoted by () which is a subgroup generated by all α-commutators. In 2016, an α-perfect group, a group that is equal to its α-commutator subgroup, was introduced by authors of this paper and the properties of such group was investigated. They proved some results on α-perfect abelian groups and showed that a cyclic group of even order is not α- perfect for any ∈ (). In this paper, we may continue our investigation on α-perfect groups and in addition to studying the relative perfectness of some classes of finite -groups, we provide an example of a non-abelian α-perfect 2-group.

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