Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G | |
986 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)(7,9)(10,11,12,13)(14,15)(16,22)(17,26)(18,20)(19,24)(21,27)(23,28)(25,29);(14,16,18,25)(15,19,20,28)(17,23,27,24)(21,29,26,22);(6,7,8,9)(15,20)(19,28)(21,26)(22,29);(15,20)(16,23)(17,27)(19,22)(24,25)(28,29)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15671);