Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C26×C4 |
| direct product | C5×#20347 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C22×D4 |
| prime radical | C25×C5 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C24×C4×C5 ◁ G | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12,13)(14,15)(16,24)(17,21)(18,22)(19,23)(20,26)(25,29)(27,28);(7,9)(14,16,18,27,19,25,17,26)(15,20,21,29,23,28,22,24);(16,25)(20,28)(24,29)(26,27);(14,17,19,18)(15,21,23,22)(16,26,25,27)(20,29,28,24)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13335);