Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21017 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1692 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)(6,7,11,17,12,16,10,15)(8,13,19,27,20,28,24,30)(9,21,22,31,23,14,25,29)(18,26,34,37,33,35,32,36);(6,8,11,24,12,20,10,19)(7,13,15,27,16,28,17,30)(9,18,25,34,23,33,22,32)(14,26,29,36,21,35,31,37);(6,9)(7,14)(8,18)(10,22)(11,25)(12,23)(13,26)(15,29)(16,21)(17,31)(19,32)(20,33)(24,34)(27,36)(28,35)(30,37);(6,10,12,11)(7,15,16,17)(8,19,20,24)(9,22,23,25)(13,27,28,30)(14,29,21,31)(18,32,33,34)(26,36,35,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14005);