Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13173 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1942 |
| prime radical | C23×C5 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,10,18,13,19,12,16)(8,20,24,37,25,42,22,14)(9,26,28,58,31,59,30,57)(11,33,32,44,34,43,27,17)(15,38,40,61,41,60,39,29)(21,47,51,66,52,67,49,56)(23,53,54,65,55,36,48,46)(35,50,45,63,62,69,64,68);(6,8,10,24,13,25,12,22)(7,14,18,20,19,37,16,42)(9,21,28,51,31,52,30,49)(11,23,32,54,34,55,27,48)(15,35,40,45,41,62,39,64)(17,36,33,46,44,53,43,65)(26,56,58,47,59,66,57,67)(29,50,38,63,61,69,60,68);(6,9,27,60,12,30,34,61,13,31,32,38,10,28,11,29)(7,15,33,58,18,40,44,59,19,41,43,57,16,39,17,26)(8,21,48,68,22,49,55,69,25,52,54,63,24,51,23,50)(14,35,46,47,20,45,53,66,37,62,65,67,42,64,36,56)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6161);