#9224287

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

4305 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)(3,8)(4,7)(5,6)(9,10,15,17)(11,12,19,13)(14,16,23,24)(18,20,22,21)(26,30)(28,35)(31,36)(34,39);(1,3,5,7)(2,4,6,8)(9,11)(10,12)(13,17)(14,22)(15,19)(16,21)(18,23)(20,24)(27,33)(31,36)(34,39)(38,40);(9,12,15,13)(10,11,17,19)(14,20,23,21)(16,18,24,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(2,6)(4,8)(9,13,15,12)(10,11,17,19)(14,20,23,21)(16,22,24,18);(1,2)(3,4)(5,6)(7,8)(9,14)(10,16)(11,18)(12,20)(13,21)(15,23)(17,24)(19,22)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,17)(14,22)(15,19)(16,20)(18,23)(21,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131483);