C₂×#33546 #7622857

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁵
socle	C₂²	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4161 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530053);