Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#34268 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(7,9)(10,11,13,12)(14,15)(16,17)(18,26)(19,20)(21,24)(22,27)(23,28)(25,29);(10,11,13,12)(15,20)(17,21)(22,26)(25,28);(6,7)(8,9)(10,12,13,11)(14,16)(15,21)(17,20)(18,23)(19,24)(22,28)(25,26)(27,29);(14,17,19,21)(15,16,20,24)(18,25,27,28)(22,23,26,29);(14,18,19,27)(15,22,20,26)(16,23,24,29)(17,25,21,28)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27256);