Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×#1332 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | D8 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle, prime radical | C23×C5 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4,5,7,6,8,9,10)(12,16)(13,19)(14,21)(17,26)(18,25)(20,23)(22,24);(11,12,15,18)(13,20,22,26)(14,23,21,17)(16,24,25,19)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4578);