Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20713 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #381 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#381 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1244 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)(8,18,20,31,21,13,26,28)(9,22,25,29,24,14,23,30)(19,27,33,36,34,32,35,37);(6,8)(7,13)(9,19)(10,20)(11,26)(12,21)(14,27)(15,28)(16,18)(17,31)(22,32)(23,33)(24,34)(25,35)(29,36)(30,37);(6,9)(7,14,16,22)(8,19)(10,23)(11,25)(12,24)(13,27,18,32)(15,29,17,30)(20,33)(21,34)(26,35)(28,36,31,37);(6,10,12,11)(7,15,16,17)(8,20,21,26)(9,23,24,25)(13,28,18,31)(14,29,22,30)(19,33,34,35)(27,36,32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13701);