#6808127

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

1748 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,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, 6715323);