Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26,28,33,29,32,37,31)(27,30,35,39,36,38,34,40);(5,6,7,8)(10,14)(15,21)(16,22)(23,24)(25,27,28,35,29,36,37,34)(26,30,33,39,32,38,31,40);(2,4)(6,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,31)(27,34)(28,37)(30,38)(32,33)(35,36);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,32)(30,38)(31,33)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287955);