#6513597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1931 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6420793);