Finite non-abelian 2-group has maximal class iff its abelianization has order four
Suppose is a non-abelian group of order . Then, the following are equivalent for :
- is a Maximal class group (?), i.e., its nilpotency class is .
- The abelianization of has order four. Equivalently, the abelianization of is a Klein four-group. Equivalently, the commutator subgroup of has index four in .