Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11978 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C5×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15,17,23,31,37,44,39,18,21,32,22,19,24,33,41)(16,25,27,42,34,40,30,35,28,43,26,36,29,38,45,20);(6,7)(8,11)(9,13)(10,12)(14,16)(15,20)(17,30)(18,28)(19,34)(21,35)(22,38)(23,40)(24,42)(25,39)(26,33)(27,44)(29,31)(32,45)(36,37)(41,43);(6,8,10,9)(7,11,12,13)(15,21)(20,35)(22,23)(24,37)(25,43)(36,42)(38,40)(39,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4966);