Finite p-group ($|G|=p^n$)

A finite p-group is a finite group whose order is a power of one prime number $p$.

Every nontrivial finite p-group has a nontrivial center. The conjugation action partitions the group into conjugacy classes whose noncentral sizes are positive powers of $p$; the class equation then implies $|Z(G)|\equiv |G|\equiv 0(\mathrm{mod}p)$.

The Frattini subgroup of a finite p-group is the intersection of its maximal subgroups. It equals $G^p[G,G]$.

The Frattini quotient of a finite p-group is an elementary abelian p-group. Its dimension over ${\mathbb{F}}_p$ is the minimum number of generators of $G$; in particular, it has dimension at least two when $G$ is noncyclic.