Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C5×#362 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4)(37,38,39,40,41);(5,6,9,13,10,15,25,14)(7,16,18,29,19,33,24,32)(8,20,22,12,23,30,36,31)(11,21,26,17,28,34,27,35);(5,7,10,19)(6,11,15,28)(8,21,23,34)(9,24,25,18)(12,29,31,32)(13,27,14,26)(16,20,33,30)(17,22,35,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5633);