Q₈² #558

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) = #72226610 (order 18432) · Inn(G) = C₂⁴ · Out(G) = #10596770⋊C₂
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

Q₈²

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂²	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

106 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

(9,10,12,14)(11,16,13,15);(9,11,12,13)(10,15,14,16);(1,2,4,6)(3,7,8,5)(9,13,12,11)(10,15,14,16);(1,3,4,8)(2,5,6,7)(9,14,12,10)(11,16,13,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 239);