Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13015 |
| 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 | |
490 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)(14,15,18,24,34,43,32,21,20,26,17,23,19,25,35,39)(16,27,29,45,37,42,33,41,31,38,28,40,30,44,36,22);(6,7,8,9)(15,21)(17,32)(18,35)(19,34)(22,38)(23,25)(24,43)(26,39)(27,41)(28,33)(29,36)(30,37)(40,44)(42,45);(11,13)(14,16,20,31)(15,22,26,41)(17,33,18,36)(19,37,34,30)(21,38,39,27)(23,42,24,44)(25,45,43,40)(28,35,29,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6003);