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C₂×#12976 #494398

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂⁴	C₂²×C₈
prime radical	C₂³×D₄	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#494403, C₂×#499713, C₂×#499712, …
maximal normal subgroup of	C₂×#494403, C₂×#499713, C₂×#499712, …
maximal quotient of	C₂×#152498, C₂×#152491

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401594);