#474384

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,21,22,24)(26,30)(31,39)(32,37)(33,40);(1,3,4,7)(2,5,6,8)(10,12)(11,16)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(9,12,13,10)(11,15,16,14)(18,22)(21,24)(27,34)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381580);