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C₂×#10945 #354115

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3090 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);(7,8,11,15,12,16,10,14)(9,17,18,13,20,21,19,22);(3,4,5,6)(7,9)(8,13)(10,19)(11,18)(12,20)(14,17)(15,21)(16,22);(8,14,16,15)(13,17,22,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215774, C₂×#379089, #72215966, …
maximal normal subgroup of	#72215774, C₂×#379089, #72215966, …
maximal quotient of	#72215714, #72215865, C₂×#110532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261311);