#6513606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1907 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)(11,20)(12,24)(13,18)(14,31)(15,19)(16,22)(17,27)(21,32)(23,33)(25,35)(26,28)(29,36)(30,34)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,22,15,23,13,30)(10,16,18,20,19,34,31,33)(12,25,28,39,29,21,27,38)(17,35,36,40,26,32,24,37);(2,5)(3,7)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,32)(18,36)(19,26)(20,37)(22,38)(23,25)(24,31)(30,39)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,26)(13,31)(14,18)(15,19)(17,29)(20,22)(21,35)(23,34)(24,28)(25,32)(27,36)(30,33)(37,39)(38,40);(2,5)(6,8)(9,13,15,14)(10,18,19,31)(11,22,23,30)(12,27,29,28)(16,33,34,20)(17,36,26,24)(21,38,25,39)(32,40,35,37)

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,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 6420802);