#9236467

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

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9143663);