Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,32,28,31,36,39)(27,33,35,40,34,30,37,38);(1,2,4,5)(3,6,7,8)(18,21)(19,23)(25,27,28,34)(26,30,31,33)(29,35,36,37)(32,38,39,40);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,20,23)(18,22,21,24);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(19,23)(22,24)(26,31)(27,34)(32,39)(35,37)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=dei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288269);