#6529984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499761
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

1476 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)(25,26,29,34)(27,35,36,31)(28,33,30,32)(37,40,38,39);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,37)(29,36)(30,38)(32,40)(33,39)(34,35);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(27,36)(28,30)(31,39)(33,34)(35,40);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,29,30)(26,33,34,32)(27,37,36,38)(31,39,35,40);(10,16)(14,17)(15,24)(22,23)(27,36)(31,35)(37,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6437180);