Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#20623 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #379 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#379 |
| prime radical | C2×C5×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
3644 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)(8,17)(9,21,24,14)(10,15)(11,25)(12,16)(13,19)(18,32,34,31)(20,28)(22,36,23,30)(26,29)(27,33,35,37);(6,8)(7,13)(9,18)(10,19)(11,26)(12,20)(14,27)(15,28)(16,29)(17,25)(21,35)(22,33)(23,37)(24,34)(30,32)(31,36);(6,9)(7,14)(8,18)(10,22)(11,23)(12,24)(13,27)(15,30)(16,21)(17,31)(19,33)(20,34)(25,36)(26,37)(28,32)(29,35);(6,10,12,11)(7,15,16,25)(8,19,20,26)(9,22,24,23)(13,28,29,17)(14,30,21,36)(18,33,34,37)(27,32,35,31)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13611);