#6499359

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470368
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ghi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406555);