Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8634 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #380 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,9,11,8,10,12,13)(14,15)(16,25)(17,30,32,37)(18,33)(19,23)(20,24)(21,27)(22,31,38,34)(26,35,42,45)(28,36)(29,40)(39,41,44,43);(7,10)(11,13)(14,16,18,27,20,29,19,28)(15,21,23,25,24,36,33,40)(17,26,31,41,32,42,34,43)(22,35,37,39,38,45,30,44);(7,10)(9,12)(14,17)(15,22)(16,26)(18,31)(19,34)(20,32)(21,35)(23,37)(24,38)(25,39)(27,41)(28,43)(29,42)(30,33)(36,45)(40,44)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1622);