#9210320

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

4317 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9117516);