#8976322

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8883518);