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C₂×#18040 #6345024

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6360208, C₂×#6360217, C₂×M₁₆×AGL(1,ℤ/8)
maximal normal subgroup of	C₂×#6360208, C₂×#6360217, C₂×M₁₆×AGL(1,ℤ/8)
maximal quotient of	C₂×#193646

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252220);