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C₂×#10826 #353545

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	#388
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215689, #72215774, C₂×#378811, …
maximal normal subgroup of	#72215689, #72215774, C₂×#378811, …
maximal quotient of	#72215665, #72215729, C₂×#123716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260741);