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C₄×#1490 #182218

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#18524, AGL(1,𝔽₅)×#1490
maximal normal subgroup of	C₄×#18524
maximal quotient of	C₄×#11669, AGL(1,𝔽₅)×#1490

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89414);