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C₂×#11082 #357749

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	#394
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#475763, #72215689, #72215774, …
maximal normal subgroup of	C₂×#475763, #72215689, #72215774, …
maximal quotient of	#72213342, #72213386, C₂×#123640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264945);