#6351506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^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,d^b=dh,f^b=fh,g^b=gi,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, 6258702);