Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12790 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | #432 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,20,15,19,29,34,16,18,28,33,13,26,27,39)(12,21,23,36,24,31,38,40,25,35,37,41,22,17,30,32);(6,7,8,9)(10,12,13,22,16,25,15,24)(11,17,19,21,18,31,26,35)(14,23,27,30,28,37,29,38)(20,32,34,36,33,40,39,41);(11,18)(14,28)(17,31)(19,26)(21,35)(23,37)(27,29)(30,38)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5778);