Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#34371 |
| 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 | |
2238 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)(6,7,8,9)(10,11,14,12)(13,16,17,15)(19,24)(20,21)(22,30)(32,33);(11,12)(15,16)(18,19,23,24)(20,26,21,25)(22,27,30,31)(28,33,29,32);(18,20,23,21)(19,25,24,26)(22,28,30,29)(27,32,31,33);(10,12,14,11)(13,16,17,15)(18,21,23,20)(19,26,24,25)(22,29,30,28)(27,33,31,32);(10,13)(11,15)(12,16)(14,17)(18,22)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(26,33)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27359);