Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#11618 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C5×#121 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
324 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)(6,7,10,13,11,8,9,12)(14,15,18,23,19,24,33,41,20,25,34,42,17,22,31,39)(16,26,29,43,28,36,44,40,27,38,32,45,30,21,35,37);(6,8)(7,11)(9,13)(10,12)(14,16)(15,21)(17,30)(18,32)(19,28)(20,27)(22,38)(23,40)(24,26)(25,36)(29,34)(31,44)(33,35)(37,42)(39,43)(41,45);(6,9,11,10)(7,12,8,13)(16,27)(21,36)(26,38)(28,30)(29,32)(35,44)(37,40)(43,45)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4606);