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×#34289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#547 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
3032 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)(6,7,9,8)(11,13)(14,15)(16,22)(17,25)(18,19)(20,24)(21,26)(23,27)(28,29);(15,19)(20,22)(21,25)(27,28);(10,11)(12,13)(14,16,18,24)(15,20,19,22)(17,23,26,29)(21,27,25,28);(14,15)(16,20)(17,21)(18,19)(22,24)(23,27)(25,26)(28,29);(6,8,9,7)(14,17)(15,21)(16,23)(18,26)(19,25)(20,27)(22,28)(24,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27277);