Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#21751 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5×Q8 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15)(16,20)(17,27,29,21)(18,22)(19,23)(24,38,40,32)(25,33)(26,34)(28,41,43,35)(30,36)(31,37)(39,45,42,44);(6,8)(7,10)(9,13)(11,12)(14,16,19,26)(15,20,23,34)(17,24,29,40)(18,30,31,25)(21,32,27,38)(22,36,37,33)(28,42,43,39)(35,45,41,44);(8,13)(10,12)(14,17)(15,21)(16,24)(18,28)(19,29)(20,32)(22,35)(23,27)(25,39)(26,40)(30,42)(31,43)(33,44)(34,38)(36,45)(37,41);(7,11)(10,12)(14,18,19,31)(15,22,23,37)(16,25,26,30)(17,28,29,43)(20,33,34,36)(21,35,27,41)(24,39,40,42)(32,44,38,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14739);