Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×C8×#137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
714 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)(6,7,8,9,10,11,12,13);(14,15)(16,22)(17,25)(18,20)(19,23)(21,27)(24,29)(26,28);(14,16,18,24)(15,19,20,28)(17,26,27,23)(21,29,25,22);(15,20)(19,28)(21,25)(22,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10839);