#6353500

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

3015 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)(9,10,16,13)(11,21,26,34)(12,27,30,33)(14,19,15,20)(17,24,23,25)(18,31,28,29)(22,32,39,36)(35,38,40,37);(3,7)(5,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,25)(16,26)(18,32)(19,21)(20,34)(27,40)(28,36)(29,37)(30,39)(31,38)(33,35);(1,2)(3,8)(4,6)(5,7)(9,12)(10,18)(11,22)(13,28)(14,29)(15,31)(16,30)(17,32)(19,33)(20,27)(21,35)(23,36)(24,37)(25,38)(26,39)(34,40);(9,13,16,10)(11,23,26,17)(12,28,30,18)(14,19,15,20)(21,25,34,24)(22,36,39,32)(27,29,33,31)(35,38,40,37);(9,14,16,15)(10,19,13,20)(11,24,26,25)(12,29,30,31)(17,21,23,34)(18,33,28,27)(22,37,39,38)(32,35,36,40)

Polycyclic presentation:

2^9|a^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, 6260696);