#6355531

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1757 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)(9,10,15,20)(11,16,22,30)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,35,36,40)(23,32,24,31)(29,38,39,37);(1,2,3,5,4,6,7,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,29)(18,31)(19,32)(20,30)(25,35)(26,37)(27,36)(28,38)(33,39)(34,40);(9,12,15,27)(10,17,20,33)(11,21,22,36)(13,26,14,28)(16,29,30,39)(18,25,19,34)(23,37,24,38)(31,35,32,40);(11,22)(16,30)(21,36)(23,24)(29,39)(31,32)(35,40)(37,38);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,22,24)(12,26,27,28)(16,31,30,32)(17,25,33,34)(21,37,36,38)(29,35,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^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,e^b=eh,g^c=gi

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, 6262727);