Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
580 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)(14,15,20,25)(16,26,29,43)(17,24,19,22)(18,31,34,39)(21,36,38,28)(23,30,41,33)(27,44,35,40)(32,42,45,37);(7,10)(11,13);(6,8,9,12)(7,11,10,13)(14,16)(15,21)(17,27)(18,32)(19,35)(20,29)(22,37)(23,40)(24,42)(25,38)(26,31)(28,33)(30,36)(34,45)(39,43)(41,44);(14,17,20,19)(15,22,25,24)(16,27,29,35)(18,30,34,33)(21,37,38,42)(23,39,41,31)(26,40,43,44)(28,32,36,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26637);