Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12649 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #442 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1762 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,18,21,34)(9,15)(10,26)(11,28)(12,17)(13,20,30,35)(14,25)(16,27)(19,31,23,37)(22,29,36,32)(24,33);(6,8,10,22,27,19,24,35,12,21,25,36,11,23,9,20)(7,13,15,31,28,29,14,18,17,30,33,37,16,32,26,34);(8,19,21,23)(9,24)(10,25)(13,29,30,32)(14,26)(15,33)(18,31,34,37)(20,36,35,22)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5637);