#6353474

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

2975 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)(12,24)(13,17)(14,18)(15,19)(16,20)(22,35)(23,27)(25,31)(26,36)(28,32)(29,33)(30,34)(37,39)(38,40);(3,7)(5,8)(9,11)(10,12)(13,22)(14,23)(15,24)(16,25)(17,26)(18,21)(19,31)(20,27)(28,36)(29,38)(30,37)(32,39)(33,35)(34,40);(1,2)(3,8)(4,6)(5,7)(9,12)(10,11)(13,26)(14,21)(15,31)(16,27)(17,22)(18,23)(19,24)(20,25)(28,39)(29,35)(30,40)(32,36)(33,38)(34,37);(9,13,16,30)(10,17,20,34)(11,22,25,37)(12,26,27,40)(14,29,15,28)(18,33,19,32)(21,35,31,39)(23,38,24,36);(9,14,16,15)(10,18,20,19)(11,23,25,24)(12,21,27,31)(13,28,30,29)(17,32,34,33)(22,36,37,38)(26,39,40,35)

Polycyclic presentation:

2^9|d^2=h,e^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, 6260670);