Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12600 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,13,12)(14,15)(16,20)(17,22)(18,23)(19,21)(24,31)(25,32)(26,30)(27,35)(28,33)(29,34)(36,42)(37,40)(38,41)(39,43)(44,45);(7,9)(14,16,17,24,18,25,27,36,19,26,28,37,29,38,39,44)(15,20,22,31,23,32,35,42,21,30,33,40,34,41,43,45);(10,12,13,11)(15,21)(20,30)(22,33)(23,34)(31,40)(32,41)(35,43)(42,45)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5588);