Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7900 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C23×C5 | C2×C22⋊C4 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,16)(12,14,17,13)(18,19,22,25,23,26,21,31)(20,27,28,33,30,32,29,24);(7,9)(10,12,14,16,15,17,13,11)(18,20)(19,24,26,33)(21,29)(22,28)(23,30)(25,27,31,32);(10,13,15,14)(11,12,16,17)(18,21,23,22)(19,25,26,31)(20,28,30,29)(24,32,33,27)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 888);