Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13265 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
338 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)(8,12)(9,10)(11,13)(14,15,18,23,19,24,33,41,20,25,34,42,17,22,27,36)(16,26,29,43,28,38,44,40,30,39,32,45,31,21,35,37);(7,10)(11,12)(14,16,27,44,17,31,34,29,20,30,33,35,19,28,18,32)(15,21,36,43,22,39,42,37,25,38,41,45,24,26,23,40);(6,8,9,13)(7,11,10,12)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6253);