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C₂×#21738 #6374103

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹¹

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6374111, #72310243, #72310244, …
maximal normal subgroup of	C₂×#6374111, #72310243, #72310244, …
maximal quotient of	C₂×#359422, #72314523, #72317250

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281299);