#6353478

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

3091 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)(11,23)(12,29)(13,19)(14,20)(15,21)(16,22)(17,27)(18,30)(24,34)(25,37)(26,36)(28,40)(31,38)(32,39)(33,35);(3,7)(5,8)(9,11)(10,17)(12,24)(13,25)(14,26)(15,27)(16,28)(18,34)(19,35)(20,23)(21,40)(22,36)(29,38)(30,32)(31,33)(37,39);(1,2)(3,8)(4,6)(5,7)(9,12,16,25)(10,18,22,35)(11,24,28,13)(14,30,15,33)(17,34,36,19)(20,37,21,29)(23,39,40,38)(26,32,27,31);(9,13,16,24)(10,19,22,34)(11,25,28,12)(14,32,15,31)(17,35,36,18)(20,39,21,38)(23,37,40,29)(26,30,27,33);(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,30,25,33)(13,31,24,32)(17,23,36,40)(18,37,35,29)(19,38,34,39)

Polycyclic presentation:

2^9|c^2=h,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, 6260674);