Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21784 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1278 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)(9,20,22,32)(10,15)(11,14)(13,21,29,24)(17,33)(18,31,23,27)(25,30)(26,34)(28,36)(35,37);(6,8)(7,12)(9,17)(10,23)(11,19)(13,28)(14,16)(15,31)(18,25)(20,37)(21,36)(22,35)(24,34)(26,29)(27,30)(32,33);(6,9,11,22)(7,13,14,29)(8,17,19,35)(10,24,25,21)(12,26,16,28)(15,32,30,20)(18,36,23,34)(27,33,31,37);(6,8)(7,12)(9,17)(10,18)(11,19)(13,26)(14,16)(15,27)(20,37)(21,34)(22,35)(23,25)(24,36)(28,29)(30,31)(32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14772);