#6351565

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3088 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)(14,19)(15,20)(16,24)(17,28)(22,30)(23,32)(25,31)(27,33)(29,34)(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)(20,32)(26,39)(27,36)(28,37)(29,38)(33,40)(34,35);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,30)(18,34)(19,26)(20,33)(21,35)(23,37)(24,36)(25,38)(31,39)(32,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,32,31)(17,26,33,34)(22,36,38,37)(30,39,40,35);(9,13,15,14)(10,19,20,18)(11,23,25,24)(12,28,29,27)(16,31,32,21)(17,26,33,34)(22,37,38,36)(30,39,40,35)

Polycyclic presentation:

2^9|d^2=h,e^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, 6258761);