#6683267

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

1893 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)(9,10,14,18)(11,22,24,16)(12,27,30,33)(13,19,34,39)(15,21,28,20)(17,32,38,31)(23,36,40,35)(25,37,26,29);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,16,18,22)(12,19,30,39)(13,33,34,27)(15,25,28,26)(17,36,38,35)(20,37,21,29)(23,31,40,32);(9,12,28,32,14,30,15,31)(10,17,21,33,18,38,20,27)(11,23,26,39,24,40,25,19)(13,29,36,22,34,37,35,16);(2,6)(5,7)(10,18)(11,24)(17,38)(19,39)(20,21)(23,40)(25,26)(27,33);(3,8)(5,7)(9,13)(10,19)(11,17)(12,29)(14,34)(15,35)(16,31)(18,39)(20,40)(21,23)(22,32)(24,38)(25,27)(26,33)(28,36)(30,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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, 6590463);