Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23535 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#394 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1352 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)(8,17)(9,22)(10,27)(11,15)(12,16)(13,19)(14,25)(18,28)(20,30)(21,29)(23,31)(24,35)(26,32)(33,36)(34,37);(6,8)(7,13)(9,23)(10,20)(11,19)(12,21)(14,31)(15,29)(16,30)(17,27)(18,26)(22,28)(24,34)(25,33)(32,37)(35,36);(6,9,10,25,12,24,11,26)(7,14,15,22,16,32,27,35)(8,18,19,34,21,33,20,23)(13,28,17,31,30,36,29,37);(9,24)(14,32)(18,33)(22,35)(23,34)(25,26)(28,36)(31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16523);