Classification of finite 2-groups of maximal class

This article gives a classification statement for certain kinds of groups of prime power order, subject to additional constraints.
View other such statements

Statement

Let $G$ be a group of order $2^n$ and nilpotency class $n - 1$, where $n \ge 4$. In other words, $G$ is a 2-group that is also a maximal class group. Then, $G$ has a cyclic maximal subgroup $M \cong \mathbb{Z}/2^{n-1}\mathbb{Z}$, and it is one of the following groups:

1. A dihedral group: it is a semidirect product of $M$ and a cyclic group of order two, which acts on $M$ via multiplication by -1.
2. A semidihedral group: a semidirect product of $M$ and a cyclic group of order two, which acts on $M$ via multiplication by $2^{n-2} - 1$.
3. A generalized quaternion group.

Note that in the case $n = 3$, we only get the dihedral group:D8 and the quaternion group, and no semidihedral group.

Particular cases

$n$	$n - 1$ (equals nilpotency class)	$2^n$	dihedral group of order $2^n$ (this is the only capable group among the three)	semidihedral group of order $2^n$	generalized quaternion group of order $2^n$
3	2	8	dihedral group:D8	--	quaternion group
4	3	16	dihedral group:D16	semidihedral group:SD16	generalized quaternion group:Q16
5	4	32	dihedral group:D32	semidihedral group:SD32	generalized quaternion group:Q32
6	5	64	dihedral group:D64	semidihedral group:SD64	generalized quaternion group:Q64

Related facts

• Finite 2-groups of same order and maximal class are isoclinic
• Classification of finite p-groups with cyclic maximal subgroup
• Classification of finite 3-groups of maximal class
• Finite non-abelian 2-group has maximal class iff its abelianization has order four

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}