Extraspecial groups

An extraspecial group is a non-abelian p-group as small as possible above its center: the center, the commutator subgroup, and the Frattini subgroup all coincide and equal a single cyclic group of order p. For each prime p and each n ≥ 1 there are exactly two, of order p¹⁺²ⁿ.

Definition

A finite p-group G is extraspecial when it is non-abelian and

Z(G) = [G, G] = Φ(G) ≅ Cₚ.

Consequences:

G/Z(G) is elementary abelian — a vector space over the field 𝔽ₚ of p elements — of even rank 2n, so |G| = p¹⁺²ⁿ.
Nilpotency class 2: [G, G] = Z(G) is central, so G is non-abelian while G/Z(G) is abelian.
Exponent p or p² (p odd); exponent 4 (p = 2).

The name is literal: an extraspecial group is a special p-group (Z(G) = [G, G] = Φ(G)) whose center is as small as it can be, Cₚ. The smallest examples have order p³ — and every non-abelian group of order p³ is extraspecial. Being non-abelian alone is not enough, though: D₄ × C₂ (order 16) has center C₂ × C₂, strictly larger than its commutator subgroup [G, G] ≅ C₂, so the three subgroups fail to coincide and it is not extraspecial.

The two types

For each p and n there are exactly two extraspecial groups of order p¹⁺²ⁿ up to isomorphism, written in ATLAS notation

p₊¹⁺²ⁿ (plus type) and p₋¹⁺²ⁿ (minus type).

Both are central products of n extraspecial groups of order p³ — the direct product of the n factors with all n copies of the order-p center identified to one shared Cₚ (written ∘). Identifying n centers into one removes n − 1 factors of p, giving order p³ⁿ / pⁿ⁻¹ = p¹⁺²ⁿ. The ± sign is an isomorphism invariant: the plus and minus groups are never isomorphic.

p = 2

The two extraspecial groups of order 8 are the smallest non-abelian 2-groups:

Group	Type	Involutions
`D₄` — dihedral	`2₊¹⁺²`	many
`Q₈` — quaternion	`2₋¹⁺²`	unique

The higher groups are central products of these:

2₊¹⁺²ⁿ  =  D₄ ∘ D₄ ∘ ⋯ ∘ D₄      (n copies of D₄)
2₋¹⁺²ⁿ  =  Q₈ ∘ D₄ ∘ ⋯ ∘ D₄      (one Q₈, n−1 copies of D₄)

An even number of Q₈ factors gives the plus type, an odd number the minus type (Q₈ ∘ Q₈ ≅ D₄ ∘ D₄). Both types have exponent 4, so exponent does not separate them: the plus type is distinguished by having strictly more involutions — exactly the difference between D₄ and Q₈.

p odd

The two extraspecial groups of order p³ are separated by exponent:

p₊¹⁺² has exponent p — the Heisenberg group, the 3 × 3 upper-unitriangular matrices over 𝔽ₚ:

$(\begin{matrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix})$ a, b, c ∈ 𝔽ₚ
p₋¹⁺² has exponent p².

In general p₊¹⁺²ⁿ has exponent p (the central product of n Heisenberg groups) and p₋¹⁺²ⁿ has exponent p² (one exponent-p² factor with n − 1 Heisenberg factors).

Symplectic structure

Write V = G/Z(G) ≅ 𝔽ₚ²ⁿ and identify Z(G) ≅ 𝔽ₚ. Because G has class 2, the commutator depends only on cosets and descends to

V × V → 𝔽ₚ,    (x̄, ȳ) ↦ [x, y],

a non-degenerate alternating (symplectic) form — non-degeneracy forces dim V = 2n. Every extraspecial group of order p¹⁺²ⁿ carries the same symplectic space; the type is recorded by a quadratic form q : V → 𝔽ₚ refining it:

p = 2: q(x̄) = x² (well-defined into Z ≅ 𝔽₂); its polarization q(x̄ + ȳ) − q(x̄) − q(ȳ) = [x, y] is the commutator form. The type ε = ± is the type (Arf invariant) of q — + for a form of O⁺ type, − for O⁻.
p odd: the p-power map x ↦ xᵖ plays the same role, vanishing identically exactly for the exponent-p (plus) type.

Z(G) is characteristic, so every automorphism acts 𝔽ₚ-linearly on V preserving q (up to a scalar). Thus the outer automorphisms act through the orthogonal group of q; for p = 2 this is exact —

Inn(G) ≅ V,     Out(G) ≅ O^ε(2n, 2),

the full orthogonal group of the quadratic form. (For example Out(D₄) ≅ O⁺(2, 2) has order 2, while Out(Q₈) ≅ O⁻(2, 2) ≅ S₃.)

Examples

Order 8 — D₄ = 2₊¹⁺² and Q₈ = 2₋¹⁺², the only two non-abelian groups of the order.
Order 27 — the Heisenberg group 3₊¹⁺² (exponent 3, unitriangular 3 × 3 over 𝔽₃) and its exponent-9 partner 3₋¹⁺²; these are the two non-abelian groups of the order.
Order 32 — 2₊¹⁺⁴ = D₄ ∘ D₄ and 2₋¹⁺⁴ = D₄ ∘ Q₈, the two extraspecial groups of the order.

Properties

The outer automorphism group of an extraspecial 2-group is an orthogonal group. In a group of order 2¹⁺²ⁿ with Z(G) = [G, G] ≅ C₂, squaring is a quadratic form q on V = G/Z(G) ≅ 𝔽₂²ⁿ whose polarization is the commutator. Every automorphism preserves q, and those acting trivially on V are exactly the inner ones, so Out(G) ≅ O^ε(2n, 2) — where ε = ± is the type of q, the same sign the group is named by.
The extraspecial groups are where O and SO stop differing. Out(2_ε¹⁺²ⁿ) ≅ O^ε(2n, 2), and over 𝔽₂ an orthogonal map's determinant is ±1 = 1 — so O and SO name one group there and the determinant separates nothing. The genuine index-2 subgroup Ω^ε(2n, 2) is cut out instead by the Dickson invariant, which replaces the determinant in characteristic 2.
An extraspecial group can have more outer automorphisms than elements. The two of order 32 are the smallest examples: Out(2₊¹⁺⁴) ≅ O⁺(4, 2) has order 72 and Out(2₋¹⁺⁴) ≅ O⁻(4, 2) ≅ S₅ has order 120, against a group of 32 elements.
Every invertible 2 × 2 matrix over 𝔽₃ is an outer automorphism of the Heisenberg group of order 27. In that group — the extraspecial 3₊¹⁺², the upper-unitriangular 3 × 3 matrices over 𝔽₃ — the commutator is the area form on V = G/Z(G) ≅ 𝔽₃². An automorphism need only preserve that form up to a scalar, and every invertible map scales it by its determinant, so Out(G) ≅ GL(2,𝔽₃), of order 48.