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×#33630 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
542 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)(6,7,9,11)(8,12,13,10)(14,15,18,21)(16,26,28,43)(17,25,20,23)(19,32,31,40)(22,38,36,29)(24,30,37,34)(27,44,35,41)(33,42,45,39);(6,8,9,13)(7,10,11,12)(15,21)(22,36)(23,25)(24,37)(26,43)(32,40)(39,42)(41,44);(14,16)(15,22)(17,27)(18,28)(19,33)(20,35)(21,36)(23,39)(24,41)(25,42)(26,32)(29,34)(30,38)(31,45)(37,44)(40,43);(14,17,18,20)(15,23,21,25)(16,27,28,35)(19,30,31,34)(22,39,36,42)(24,40,37,32)(26,41,43,44)(29,33,38,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26618);