Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#34179 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(6,7,8,9)(12,16)(14,17)(18,19,23,21)(20,24,28,26)(22,29,30,25)(27,32,33,31);(10,11,13,15)(12,14,16,17)(19,21)(24,26)(25,29)(31,32);(10,12)(11,14)(13,16)(15,17)(18,20,23,28)(19,24,21,26)(22,27,30,33)(25,31,29,32);(18,21,23,19)(20,26,28,24)(22,29,30,25)(27,32,33,31);(18,22)(19,25)(20,27)(21,29)(23,30)(24,31)(26,32)(28,33)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27167);