#6685506

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

1835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=f,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, 6592702);