Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7451 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | #357 |
| commutator | C2×C16 | C2×C4×C5 |
| Frattini, cosocle | C4×C16 | C22×C5 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,17,23,33,42,45,21,20,26,16,27,19,25,34,39,18,24,31,38,29,40,28,37,36,43,30,22,35,41,32,44);(6,7,9,11)(8,13,12,10)(15,21)(16,28)(17,32)(19,29)(20,36)(22,25)(23,42)(24,37)(26,44)(27,41)(30,45)(31,34)(33,35)(38,40)(39,43)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 439);