#6351670

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351670
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

3092 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)(12,25)(13,20)(14,19)(15,18)(16,24)(17,27)(22,30)(23,33)(26,34)(28,31)(29,32)(35,38)(36,40)(37,39);(3,7)(5,8)(9,11)(10,16)(12,22)(13,28)(14,24)(15,23)(17,30)(18,31)(19,33)(20,21)(25,39)(26,36)(27,38)(29,37)(32,40)(34,35);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,26)(16,30)(18,32)(19,25)(20,34)(21,35)(23,36)(24,37)(28,38)(31,40)(33,39);(12,26)(17,32)(22,36)(25,34)(27,29)(30,40)(35,39)(37,38);(10,18)(11,23)(17,32)(19,20)(22,36)(24,28)(25,34)(37,38)

Polycyclic presentation:

2^9|f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,e^b=eh,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, 6258866);