#6811500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1971 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)(3,6)(4,8)(5,7)(9,10)(11,16)(12,25)(13,18)(14,20)(15,21)(17,24)(19,26)(22,31)(23,32)(27,40)(28,34)(29,36)(30,37)(33,39)(35,38);(1,3,6,2)(4,8,5,7)(9,11,15,24)(10,16,21,17)(12,26,23,14)(13,28,30,27)(18,34,37,40)(19,32,20,25)(22,38,39,29)(31,35,33,36);(9,12,14,11,15,23,26,24)(10,17,20,25,21,16,19,32)(13,27,29,22,30,28,38,39)(18,33,36,40,37,31,35,34);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,13,26,38,15,30,14,29)(10,18,19,35,21,37,20,36)(11,22,12,27,24,39,23,28)(16,31,25,40,17,33,32,34)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=f,e^2=fhi,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,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718696);