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C₂×#11340 #358484

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2);(4,6)(7,8,9,10)(11,12,15,13)(14,18,16,17);(11,13,14,18,15,12,16,17);(3,4,5,6)(11,12)(13,15)(14,17)(16,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214149, C₂×#11711337, #72215680, …
maximal normal subgroup of	#72214149, #72215680, C₂×#379083
maximal quotient of	#72214091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265680);