Finite Group Databasebeta

← Prev (#493178) Next (#493180) →

C₄×#2165 #493179

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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);(5,6,7,12,8,13,15,20,9,10,14,17,16,11,18,19);(6,10)(7,14)(11,13)(15,18);(6,11,10,13)(7,14)(12,20,17,19)(15,18)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#2165, #72224852, #72224853, …
maximal normal subgroup of	#72224852, #72224853, C₄×#33688, …
maximal quotient of	AGL(1,𝔽₅)×#2165, C₄×#12607, C₄×#12659

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400375);