#6350559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2961 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257755);