#9221180

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

4365 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=h,e^2=g,f^2=g,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, 9128376);