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C₄×#2200 #495748

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
socle	C₂²	C₂×B(4,𝔽₂)
prime radical	C₂×#2200	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#33551, C₄×#4626⋊C₂, AGL(1,𝔽₅)×#2200, …
maximal normal subgroup of	C₄×#33551, #72278320, #72278370, …
maximal quotient of	AGL(1,𝔽₅)×#2200, #72283102, #72317819

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402944);