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×C4 |
| direct product | C5×#13244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,3,4,5)(7,9)(10,11,16,23,18,24,17,42)(12,25,29,45,30,53,41,56)(13,31,34,63,36,64,35,50)(14,22,15,40,39,54,38,21)(19,43,46,27,48,28,47,37)(20,49,51,66,52,32,62,33)(26,44,59,69,60,71,67,70)(55,65,61,68,72,57,73,58);(6,7,8,9)(10,12,14,37,17,29,38,43,18,30,39,27,16,41,15,28)(11,19,40,25,23,47,54,56,24,48,21,53,42,46,22,45)(13,26,32,65,35,59,66,68,36,60,49,57,34,67,33,58)(20,44,63,55,51,70,64,73,52,71,50,72,62,69,31,61);(10,13,14,32,17,35,38,66,18,36,39,49,16,34,15,33)(11,20,40,63,23,51,54,64,24,52,21,50,42,62,22,31)(12,26,37,65,29,59,43,68,30,60,27,57,41,67,28,58)(19,44,25,55,47,70,56,73,48,71,53,72,46,69,45,61)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6232);