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C₂×#7176 #126780

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#340
prime radical	C₂⁴	#103
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#129171, C₂×#128557, C₂×C₈×#361
maximal normal subgroup of	C₂×#129171, C₂×#128557, C₂×C₈×#361
maximal quotient of	C₂×#93270

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33976);