Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2541 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(5,6)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24);(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,39,37);(1,3)(2,5)(4,8)(6,7)(25,27,31,38)(26,32,35,36)(28,29,34,39)(30,37,40,33);(1,2,4,7)(3,5,8,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,28)(26,30)(27,29)(31,34)(32,37)(33,36)(35,40)(38,39);(1,2)(3,5)(4,7)(6,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23)(25,29)(26,33)(27,34)(28,38)(30,32)(31,39)(35,37)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30)(26,34)(27,37)(28,35)(29,36)(31,40)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=h,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=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9128390);