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C₂×#7173 #126774

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

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	#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)(12,17,18,19)(13,21)(14,23)(15,22)(20,26,24,25);(4,8,10,9)(11,12,14,19,15,20,23,26,16,18,13,17,22,24,21,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₈×#365, C₂×#128692, C₂×#128557, …
maximal normal subgroup of	C₂×C₈×#365, C₂×#128692, C₂×#128557, …
maximal quotient of	C₂×#93269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33970);