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C₄×#1972 #354584

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂²×D₄
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#21690, C₄×#13025, C₄×#13138, …
maximal normal subgroup of	C₄×#21690, C₄×#13025, C₄×#13138, …
maximal quotient of	C₄×#9837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261780);