Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7975 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#95 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C42×C5 ◁ G | |
590 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)(7,12,14,25)(8,16)(9,20)(13,23,24,35)(15,29,28,36)(18,32)(19,31)(21,22)(26,27,37,33)(30,34);(6,7,9,15,11,14,22,28)(8,12,18,26,19,25,32,37)(10,13,21,27,17,24,20,33)(16,23,30,36,31,35,34,29);(7,13)(12,23)(14,24)(15,27)(25,35)(26,36)(28,33)(29,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 963);