C₂×Q₈×C₄⋊Q₈ #7630262

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

C₂×Q₈×C₄⋊Q₈

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8082 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

(1,2);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10);(12,14)(15,16,19,22)(17,21,18,20);(15,17,19,18)(16,20,22,21);(11,12,13,14)(15,18,19,17)(16,21,22,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537458);