#6353477

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
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

3017 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)(10,16)(11,20)(13,14)(18,29)(19,26)(21,34)(22,27)(24,25)(28,31)(33,36)(35,37)(38,40);(3,7)(5,8)(9,10)(11,17)(12,18)(13,26)(14,16)(15,19)(20,31)(21,30)(22,32)(23,33)(24,36)(25,29)(27,28)(34,40)(35,39)(37,38);(1,2)(3,8)(4,6)(5,7)(9,11,15,22)(10,17,19,32)(12,21,23,35)(13,20,14,27)(16,28,26,31)(18,30,33,39)(24,34,25,37)(29,38,36,40);(9,12)(10,18)(11,21)(13,24)(14,25)(15,23)(16,29)(17,30)(19,33)(20,34)(22,35)(26,36)(27,37)(28,38)(31,40)(32,39);(12,23)(18,33)(21,35)(24,25)(29,36)(30,39)(34,37)(38,40)

Polycyclic presentation:

2^9|c^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh

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, 6260673);