Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#22429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
1466 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)(17,25)(18,20)(19,24)(21,27)(23,29)(26,28);(6,7)(8,9)(16,23)(19,28)(22,29)(24,26);(14,16,18,23)(15,19,20,28)(17,26,27,24)(21,29,25,22);(7,9)(15,20)(19,28)(21,25)(22,29)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15417);