Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#21431 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | C23 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ G | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21)(22,23,27,24,26,29,25,28);(7,10)(11,13)(22,24,27,29,26,28,25,23);(6,8,9,12)(7,11,10,13)(15,18)(19,21)(22,25,26,27)(23,24,29,28);(14,16,17,20)(15,19,18,21)(22,25,26,27)(23,28,29,24)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14419);