Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#33495 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,18,23,19,24,30,38,17,22,28,36,29,37,41,45)(16,21,26,34,27,35,40,44,25,20,31,32,39,33,42,43);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,25)(18,26)(19,27)(21,22)(23,32)(24,33)(28,31)(29,39)(30,40)(34,36)(35,37)(38,43)(41,42)(44,45);(8,13)(10,12)(14,16)(15,21)(17,25)(18,26)(19,27)(20,22)(23,34)(24,35)(28,31)(29,39)(30,40)(32,36)(33,37)(38,44)(41,42)(43,45);(7,11)(10,12)(15,22)(16,25)(23,36)(24,37)(26,31)(27,39)(38,45)(40,42)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26483);