Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24474 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1298 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)(6,7,11,16,12,10,29,15)(8,17,21,19,22,31,20,13)(9,23,27,25,28,33,26,14)(18,30,35,37,36,34,24,32);(6,8)(7,13)(9,24)(10,19)(11,21)(12,22)(14,32)(15,31)(16,17)(18,27)(20,29)(23,30)(25,37)(26,36)(28,35)(33,34);(6,9,12,28)(7,14,10,25)(8,18,22,36)(11,26,29,27)(13,30,19,34)(15,23,16,33)(17,32,31,37)(20,35,21,24);(6,10)(7,12)(8,19)(9,25)(11,15)(13,22)(14,28)(16,29)(17,20)(18,34)(21,31)(23,26)(24,37)(27,33)(30,36)(32,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17462);