#9236470

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
prime radical	#58890	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 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)(3,7)(4,6)(5,8)(9,10,15,12)(11,18,14,16)(13,17,21,20)(19,24,22,23)(25,26,32,36)(27,37,39,33)(28,30,34,40)(29,31,35,38);(1,2,4,6)(3,5,8,7)(13,21)(17,20)(19,22)(23,24)(25,27,32,39)(26,33,36,37)(28,38,34,31)(29,30,35,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,28,32,34)(26,30,36,40)(27,38,39,31)(29,33,35,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,18,14,16)(13,20,21,17)(19,24,22,23)(25,29)(26,31)(27,30)(28,33)(32,35)(34,37)(36,38)(39,40);(9,13,15,21)(10,17,12,20)(11,19,14,22)(16,23,18,24)(25,30)(26,34)(27,35)(28,36)(29,39)(31,33)(32,40)(37,38);(9,14,15,11)(10,18,12,16)(13,22,21,19)(17,24,20,23)(25,31)(26,35)(27,28)(29,36)(30,37)(32,38)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=ghi,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, 9143666);