Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#12556 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | C2×D8 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
606 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)(10,11,12,13)(14,15)(16,27)(17,22)(18,24)(19,23)(20,26)(21,29)(25,33)(28,41)(30,35)(31,42)(32,39)(34,40)(36,44)(37,45)(38,43);(7,9)(14,16,19,30,17,28,32,43,20,31,34,45,33,44,18,29)(15,21,24,36,25,37,40,42,26,38,39,41,22,35,23,27);(14,17,20,33)(15,22,26,25)(16,28,31,44)(18,19,32,34)(21,35,38,37)(23,39,40,24)(27,41,42,36)(29,30,43,45)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5544);