Abstract: The research of nilpotent groups is a important part of the research of finite groups'theory. The research report of nilpotent groups is plentiful. Decribing properties of nilpotent groups is also difficult. As known， nilpotent groups G have a normal subgroups series：G = G₁…G_r= 1 which makesG_i/G_i+1 ≤Z(G/G_i+1)(i = 1,2,…,r) discusses the generalization of nilpotent groups. Assumes that the factor group is super central and G/G_i+1 is cyclic group. According to it，introduces a definition of super nilpotent groups. Discuss super nilpotent groups' properties with using the theory of finite groups，obtains some meaningful conclusions.

Key words: super center, group series, super nilpotent group, characterization subgroup