Finite non-abelian 2-group has maximal class iff its abelianization has order four

Statement

Suppose $P$ is a non-abelian group of order $2^{n}$. Then, the following are equivalent for $P$:

• $P$ is a Maximal class group (?), i.e., its nilpotency class is $n-1$.
• The abelianization of $P$ has order four. Equivalently, the abelianization of $P$ is a Klein four-group. Equivalently, the commutator subgroup of $P$ has index four in $P$.

Related facts

• Classification of finite 2-groups of maximal class

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 194, Section 5.4 (p-groups of small depth), Theorem 4.5, ^{More info}