Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1732 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)(6,7,11,16,12,17,10,15)(8,18,22,29,21,28,20,13)(9,23,24,31,26,14,25,30)(19,32,33,27,35,37,34,36);(6,8)(7,13,17,29)(9,19)(10,20)(11,22)(12,21)(14,27,23,36)(15,28,16,18)(24,33)(25,34)(26,35)(30,37,31,32);(6,9)(7,14)(8,19)(10,24)(11,25)(12,26)(13,27)(15,30)(16,31)(17,23)(18,32)(20,33)(21,35)(22,34)(28,37)(29,36);(6,10,12,11)(7,15,17,16)(8,20,21,22)(9,24,26,25)(13,28,29,18)(14,30,23,31)(19,33,35,34)(27,37,36,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14000);