#9143337

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

4431 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,20)(14,15)(18,24)(19,22)(21,23);(1,3,7,6)(2,4,8,5)(9,10)(11,16)(12,21)(13,18)(14,15)(17,23)(19,22)(20,24)(25,26,30,34)(27,32,37,31)(28,29,33,39)(35,40,38,36);(1,4,7,5)(2,3,8,6)(10,15)(11,19)(17,21)(18,20)(25,27,30,37)(26,31,34,32)(28,35,33,38)(29,36,39,40);(2,8)(4,5)(9,11,14,19)(10,16,15,22)(12,18,23,20)(13,17,24,21)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,5,7,4)(2,3,8,6)(9,12)(10,17)(11,20)(13,16)(14,23)(15,21)(18,19)(22,24)(25,28)(26,29)(27,35)(30,33)(31,36)(32,40)(34,39)(37,38);(1,6,7,3)(2,5,8,4)(9,13)(10,18)(11,17)(12,22)(14,24)(15,20)(16,23)(19,21)(25,29)(26,33)(27,36)(28,34)(30,39)(31,38)(32,35)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9050533);