Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#20422 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,19,16,20,14,29)(12,21,25,31,23,35,24,17)(13,18,28,33,26,32,27,39)(22,34,37,40,38,41,36,30);(7,9)(10,12)(11,17,20,31)(13,22)(14,24)(15,25)(16,23)(18,30,32,40)(19,21,29,35)(26,38)(27,36)(28,37)(33,34,39,41);(10,13,15,28,16,26,14,27)(11,18,19,33,20,32,29,39)(12,22,25,37,23,38,24,36)(17,30,21,34,31,40,35,41);(12,23)(13,26)(17,31)(18,32)(21,35)(24,25)(27,28)(33,39)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13410);