Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8128 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #95 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C42×C5 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(14,15,18,23,19,22,17,21)(16,24,25,29,26,20,27,28);(6,7,8,11,9,10,12,13)(14,16)(15,20)(17,25)(18,27)(19,26)(21,28)(22,24)(23,29);(14,17,19,18)(15,21,22,23)(16,25,26,27)(20,28,24,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116);