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×#34262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(11,14)(15,17)(18,19,24,22)(20,27,21,25)(23,30,31,26)(28,32,29,33);(18,20,24,21)(19,25,22,27)(23,28,31,29)(26,32,30,33);(6,8,9,7)(10,11)(12,15)(13,14)(16,17)(18,21,24,20)(19,25,22,27)(23,29,31,28)(26,32,30,33);(18,22,24,19)(20,25,21,27)(23,30,31,26)(28,32,29,33);(10,12,13,16)(11,15,14,17)(18,23)(19,26)(20,28)(21,29)(22,30)(24,31)(25,32)(27,33)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27250);