Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23472 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#393 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,13)(9,21,23,32)(10,15)(11,26)(12,16)(14,24,31,25)(17,30,33,37)(18,28)(19,29)(20,35)(22,27,34,36);(6,8)(7,13)(9,22,23,34)(10,19)(11,20)(12,18)(14,30,31,37)(15,29)(16,28)(17,25,33,24)(21,36,32,27)(26,35);(6,9)(7,14)(8,17)(10,25)(11,24)(12,23)(13,27)(15,32)(16,31)(18,33)(19,22)(20,34)(21,26)(28,36)(29,37)(30,35);(8,18)(13,28)(17,33)(19,20)(22,34)(27,36)(29,35)(30,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16460);