Finite Group Databasebeta

← Prev (#6373931) Next (#6373933) →

C₄×#3184 #6373932

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#3184	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#3184, C₄×#19745, #72224708, …
maximal normal subgroup of	C₄×#19745, #72224708, #72224719, …
maximal quotient of	AGL(1,𝔽₅)×#3184, C₄×#21690

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281128);