#6351562

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351559
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6258758);