Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13364 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(10,11,13,15)(12,16,17,14)(19,23)(20,26)(21,28)(24,33)(25,32)(27,30)(29,31);(6,7)(8,9)(10,12,13,17)(11,14,15,16);(7,9)(18,19,22,25)(20,27,29,33)(21,30,28,24)(23,31,32,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6352);