Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(11,12)(14,15,16,19)(17,20,23,27)(18,26,24,22)(21,29,28,25);(6,7)(8,9)(10,11)(12,13)(15,19)(20,27)(21,28)(22,26);(7,9)(10,12,13,11)(15,20)(17,25)(18,24)(19,27)(21,22)(23,29)(26,28)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6371);