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C₂×#9542 #202318

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂⁴	C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(4,5)(7,8,9,11)(10,12,14,13)(15,16,18,21,19,17,20,22);(3,4,6,5)(8,11)(12,13)(15,17,18,22,19,16,20,21);(3,5,6,4)(8,12)(10,14)(11,13)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#210434, C₂×#227634, C₂×#229109, …
maximal normal subgroup of	C₂×#210434, C₂×#227634, C₂×#229109, …
maximal quotient of	C₂×#140121

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109514);