Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13248 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1972 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
394 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)(8,20)(9,27,31,57)(10,33)(11,16)(12,37)(13,19)(14,23)(15,30,44,51)(17,35)(18,34)(21,46)(22,38,54,67)(24,40)(25,47)(26,50)(28,58,60,42)(29,61,63,45)(32,65,62,43)(36,49)(39,55)(41,56)(48,53,68,59)(52,66,64,69);(6,8,21,24,12,25,10,23,13,26,11,36,34,55,35,56)(7,14,17,47,18,40,16,20,19,41,33,39,37,49,46,50)(9,28,30,64,32,54,63,53,31,60,51,52,62,22,29,59)(15,42,57,68,45,67,65,69,44,58,27,48,61,38,43,66);(6,9,11,30,12,32,35,63,13,31,21,51,34,62,10,29)(7,15,17,27,18,45,16,43,19,44,33,57,37,61,46,65)(8,22,24,53,25,28,23,52,26,54,36,59,55,60,56,64)(14,38,39,66,40,42,50,68,41,67,47,69,49,58,20,48)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6236);