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C₂×#60309 #10574469

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂×C₄	C₂⁶
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×H(5,𝔽₂)	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

7938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#10574511, C₂×#6345094, C₂×#208806, …
maximal normal subgroup of	C₂×#10574511, C₂×#6345094, C₂×#208806, …
maximal quotient of	C₂×D₄×#435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481665);