Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22277 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,15)(13,16)(14,17)(18,19)(20,28)(21,32)(22,26)(23,37)(24,30)(25,35)(27,36)(29,43)(31,39)(33,47)(34,46)(38,48)(40,44)(41,49)(42,45);(6,8,9,7)(10,12,15,11)(13,16,14,17)(18,20)(19,24)(21,33)(22,30)(23,31)(25,40)(26,28)(27,39)(29,34)(32,43)(35,48)(36,45)(37,42)(38,41)(44,49)(46,47);(10,13,15,14)(11,16,12,17)(18,21,23,35,22,34,36,49)(19,25,27,32,26,41,37,46)(20,29,31,44,30,33,45,48)(24,38,39,47,28,40,42,43);(10,14,15,13)(11,16,12,17)(19,26)(20,30)(25,41)(27,37)(29,33)(31,45)(32,46)(44,48)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15265);