Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C5×#497 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
3350 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)(25,26,27,28,29);(3,4);(6,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24);(5,6,7,8)(11,18)(14,23)(17,24)(19,22)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53622);