Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12786 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
454 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)(10,11,13,18,14,19,24,31,15,17,23,29,25,30,35,39)(12,16,21,27,22,28,34,38,20,26,32,36,33,37,40,41);(6,7,8,9)(10,12)(11,16)(13,21)(14,22)(15,20)(17,26)(18,27)(19,28)(23,32)(24,34)(25,33)(29,36)(30,37)(31,38)(35,40)(39,41);(11,17)(12,20)(13,23)(19,30)(22,33)(24,35)(27,36)(38,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5774);