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C₂×#12920 #494235

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1350 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);(3,4)(5,6)(7,8,10,9)(11,12,14,18)(13,16,22,24)(15,23,21,19)(17,26,25,20);(7,9,10,8)(12,16)(13,20)(15,21)(17,19)(18,24)(22,26)(23,25);(4,6)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#495688, #72286620, #72286659, …
maximal normal subgroup of	C₂×#495688, #72286620, #72286659, …
maximal quotient of	C₂×#152427, #72286659, C₂×#152432, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401431);