Q₈ #13

order	8
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = S₄ (order 24) · Inn(G) = C₂² · Out(G) = D₃
holomorph	G ⋊ Aut(G) = #5935
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ G
LECS	G ▷ C₂ ▷ C₁

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8)

Polycyclic presentation:

2^3|a^2=c,b^2=c,b^a=bc

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8, 4);