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C₂×#7174 #126776

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#106
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#340
prime radical	C₂⁴	#102
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

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#126882, C₂×#128555, C₂×#128572, …
maximal normal subgroup of	C₂×#126882, C₂×#128555, C₂×#128572, …
maximal quotient of	C₂×#93269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33972);