#6683268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590464);