#6561590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6468786);