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) = C2×C4 |
| direct product | C5×#12881 |
| 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 | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
298 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)(7,8)(10,11,14,21,15,18,30,34,16,19,29,35,13,20,27,36)(12,22,24,37,25,32,39,41,26,17,28,33,23,31,38,40);(6,7,9,8)(10,12,13,23,16,26,15,25)(11,17,20,32,19,22,18,31)(14,28,27,39,29,24,30,38)(21,37,36,40,35,33,34,41);(6,8,9,7)(11,18,19,20)(14,29)(17,31,22,32)(21,36,35,34)(24,28)(27,30)(33,41,37,40)(38,39)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5869);