C₂×Q₈×C₂²⋊Q₈ #7630244

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

Aliases

C₂×Q₈×C₂²⋊Q₈

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

8758 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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)(15,16,18,20)(17,21,22,19);(15,17,18,22)(16,19,20,21);(11,12)(13,14)(15,16)(17,19)(18,20)(21,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537440);