Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#13190 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C22×C5×D4 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,9)(11,13)(14,15,19,23,20,22,27,28,18,16,21,25,17,24,26,29);(6,7,8,9)(14,16,19,25,20,24,27,29,18,15,21,23,17,22,26,28);(10,11,12,13)(14,17,18,20)(15,22,16,24)(19,26,21,27)(23,28,25,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6178);