Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10013 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #392 |
| prime radical | C5×#3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15)(16,22)(17,25)(18,21)(19,23)(20,27)(24,29)(26,28);(6,7,9,11)(8,13,12,10)(14,16,18,24)(15,19,21,28)(17,26,27,23)(20,29,25,22);(6,7)(8,13)(9,11)(10,12)(19,28)(22,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3001);