Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C43 |
| direct product | C5×#14638 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(6,7,8,9)(11,14)(15,17)(18,19,22,21)(20,23,24,25)(28,32)(30,33);(12,16)(15,17)(19,21)(23,25)(26,27,29,31)(28,33,32,30);(10,11)(12,15)(13,14)(16,17)(18,20)(19,23)(21,25)(22,24)(26,28,29,32)(27,30,31,33);(10,12)(11,15)(13,16)(14,17)(18,21,22,19)(20,23,24,25)(26,27)(28,30)(29,31)(32,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 7626);