Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#21372 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
714 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)(14,15,19,23,20,24,18,32)(16,25,27,21,28,35,33,36)(17,22,31,38,29,37,30,42)(26,39,41,34,40,45,43,44);(14,16,19,27,20,28,18,33)(15,21,23,35,24,36,32,25)(17,26,31,41,29,40,30,43)(22,34,38,45,37,44,42,39);(7,10)(11,13)(14,17)(15,22)(16,26)(18,30)(19,31)(20,29)(21,34)(23,38)(24,37)(25,39)(27,41)(28,40)(32,42)(33,43)(35,45)(36,44);(6,8,9,12)(7,11,10,13)(17,29)(22,37)(26,40)(30,31)(34,44)(38,42)(39,45)(41,43)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14360);