Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33678 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#2184 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
568 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)(8,11)(9,10)(12,13)(14,15)(16,20)(17,21)(18,31)(19,24)(22,28)(23,29)(25,33)(26,32)(27,41)(30,37)(34,40)(35,39)(36,42)(38,43)(44,45);(7,10)(11,13)(14,16)(15,20)(17,26)(18,27)(19,25)(21,32)(22,34)(23,35)(24,33)(28,40)(29,39)(30,38)(31,41)(36,44)(37,43)(42,45);(15,21)(17,28)(18,29)(20,32)(22,23)(24,37)(26,40)(27,39)(30,42)(31,36)(33,43)(34,35)(38,45)(41,44);(6,8,9,12)(7,11,10,13)(16,25)(20,33)(26,38)(27,39)(32,43)(34,44)(35,41)(40,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26666);