#6355306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355299
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60940	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1993 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)(3,6)(7,8)(9,10)(11,15)(12,22)(13,25)(14,17)(16,23)(18,27)(19,33)(20,29)(21,37)(24,32)(26,31)(28,35)(30,36)(34,40)(38,39);(1,2,6,8,4,7,3,5)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,29)(18,30)(22,33)(23,35)(24,34)(25,37)(26,38)(27,36)(31,39)(32,40);(9,12,14,24)(10,16,17,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(18,32,25,22)(21,38,36,35)(30,40,37,33);(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38);(10,17)(15,29)(16,31)(18,25)(22,32)(28,39)(30,37)(33,40)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262502);