Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12623 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | C4×#99 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,12,9,14,23)(8,17,20,34,21,31,28,37)(10,15,25,32,26,22,30,36)(13,27,29,24,18,33,35,19);(6,8)(7,13)(9,18)(10,24,26,19)(11,20)(12,21)(14,28)(15,31,22,17)(16,29)(23,35)(25,33,30,27)(32,37,36,34);(6,9,11,23,12,7,14,16)(8,18,20,35,21,13,28,29)(10,22,25,36,26,15,30,32)(17,33,34,19,31,27,37,24)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5611);