#6806837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
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

1965 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,15)(13,18)(14,19)(16,20)(21,25)(22,33)(23,34)(24,26)(27,31)(28,30)(29,35)(32,36)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,16,25)(10,17,20,21)(12,26,18,33)(13,22,15,24)(14,29,32,38)(19,35,36,40)(23,28,39,31)(27,34,30,37);(9,12,15,20,16,18,13,10)(11,21,22,26,25,17,24,33)(14,30,31,19,32,27,28,36)(23,34,29,35,39,37,38,40);(2,6)(5,7)(9,13,16,15)(10,18,20,12)(11,22,25,24)(14,28,32,31)(17,33,21,26)(19,30,36,27)(23,29,39,38)(34,35,37,40);(3,8)(5,7)(9,14,15,31,16,32,13,28)(10,19,12,27,20,36,18,30)(11,23,22,29,25,39,24,38)(17,34,33,35,21,37,26,40)

Polycyclic presentation:

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

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, 6714033);