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C₂×#11097 #357672

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂×C₄⋊C₄
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,6)(11,12,15,18,16,19,14,24)(13,20,22,25,23,26,21,17);(7,8)(9,10)(11,13,14,21,16,23,15,22)(12,17,18,20,19,25,24,26);(4,6)(8,10)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214941, #72215970, #72215994, …
maximal normal subgroup of	#72214941, #72215970, #72215994, …
maximal quotient of	C₂×#117897, #72214917

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264868);