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C₂×#10723 #353308

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂×C₄⋊C₄
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5106 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)(7,8)(9,10)(11,12,14,19,16,20,15,18)(13,21,22,17,24,25,23,26);(4,6)(11,13,15,23,16,24,14,22)(12,17,19,25,20,26,18,21);(8,10)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#392182, C₂×#354302, C₂×#392822, …
maximal normal subgroup of	C₂×#392182, C₂×#354302, C₂×#392822, …
maximal quotient of	#72321984

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260504);