Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13207 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11,12,13)(14,15)(16,24)(17,29)(18,21)(19,33)(20,27)(22,32)(23,30)(25,37)(26,39)(28,45)(31,34)(35,42)(36,41)(38,44)(40,43);(11,13)(14,16,25,41,19,28,17,27,18,26,32,44,30,43,31,42)(15,20,34,45,23,36,22,24,21,35,29,40,33,38,37,39);(6,7,8,9)(15,21)(16,26)(22,37)(23,33)(27,42)(28,43)(29,34)(41,44)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6195);