Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13011 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C4×D8 |
| prime radical | C2×C5×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
476 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)(10,11,16,23,17,24,14,21,18,25,34,40,26,35,15,22)(12,20,28,19,29,37,13,31,30,36,33,38,32,41,27,39);(6,7,8,9)(10,12,26,29,18,30,17,32)(11,19,35,31,25,38,24,39)(13,15,33,34,27,14,28,16)(20,21,37,23,36,22,41,40);(10,13)(11,20)(12,15)(14,30)(16,32)(17,28)(18,27)(19,21)(22,38)(23,31)(24,41)(25,36)(26,33)(29,34)(35,37)(39,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5999);