Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17477 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C2×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C5×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C22×C42×C5 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,14)(10,18)(11,13)(12,16)(17,21)(19,20)(22,23,25,29,26,28,32,35)(24,27,31,33,30,34,36,37);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(7,13)(12,20)(14,18)(15,21)(23,28)(24,30)(29,35)(31,36);(6,9,11,10)(7,14,13,18)(8,16,17,19)(12,21,20,15)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10465);