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C₂×#17891 #6344583

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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	C₂²×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2124 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)(11,12,15,13,14,16,18,17);(4,7)(5,10)(8,9)(12,16)(13,17);(11,13,15,16,14,17,18,12);(4,8)(7,9)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6361594, C₂×#6361602, C₂×M₁₆×AGL(1,ℤ/8)
maximal normal subgroup of	C₂×#6361594, C₂×#6361602, C₂×M₁₆×AGL(1,ℤ/8)
maximal quotient of	C₂×#193555

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251779);