Classification of finite 2-groups of maximal class

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

References

Textbook references