Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#106 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
318 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)(12,20,22,32)(13,24)(14,17)(15,28)(16,21,29,34)(18,26)(19,27)(23,37,36,33)(25,38)(30,41,40,35)(31,39);(10,12,13,21,15,23,26,35,14,22,25,34,27,36,39,41)(11,16,18,20,19,30,24,33,17,29,31,32,28,40,38,37);(6,7,8,9)(11,17)(16,29)(18,31)(19,28)(20,32)(24,38)(30,40)(33,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5892);