Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13260 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,19,32,20,30,36,16,22,29,18,15,21,13,28)(12,23,24,38,25,33,40,35,27,39,41,34,26,37,31,17);(6,7,8,9)(10,12)(11,17)(13,24)(14,31)(15,25)(16,27)(18,33)(19,37)(20,34)(21,38)(22,35)(23,28)(26,32)(29,40)(30,41)(36,39);(10,13,15,29,16,30,32,14)(11,18,20,28,22,19,21,36)(12,24,25,40,27,41,26,31)(17,33,34,23,35,37,38,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6248);