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×#20710 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #381 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | 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 | |
1276 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)(8,16,19,12)(9,20)(10,14)(11,15)(13,22)(17,24,31,28)(18,29,32,25)(21,33)(23,27)(26,34)(30,35,37,36);(6,8)(7,12)(9,17)(10,18)(11,19)(13,24)(14,25)(15,16)(20,28)(21,30)(22,31)(23,32)(26,35)(27,29)(33,36)(34,37);(6,9)(7,13,15,20)(8,17)(10,21)(11,22)(12,24,16,28)(14,26,27,33)(18,30)(19,31)(23,34)(25,35,29,36)(32,37);(6,10,11,23)(7,14,15,27)(8,18,19,32)(9,21,22,34)(12,25,16,29)(13,26,20,33)(17,30,31,37)(24,35,28,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13698);