Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60307 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,18,23,19,21,29,34)(16,20,26,32,24,30,37,41)(17,22,27,35,28,33,39,43)(25,31,38,42,36,40,44,45);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,25)(18,26)(19,24)(21,30)(22,31)(23,32)(27,38)(28,36)(29,37)(33,40)(34,41)(35,42)(39,44)(43,45);(8,13)(10,12)(15,21)(16,24)(22,33)(23,34)(25,36)(26,37)(35,43)(38,44);(7,11)(10,12)(16,24)(20,30)(25,36)(26,37)(31,40)(32,41)(38,44)(42,45);(14,17,18,27,19,28,29,39)(15,22,23,35,21,33,34,43)(16,25,26,38,24,36,37,44)(20,31,32,42,30,40,41,45)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53295);