Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#35018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(8,12)(11,13)(14,15,17,19)(16,18,20,21)(23,27)(28,29);(7,10)(11,13)(14,15)(16,18)(17,19)(20,21)(22,23)(24,28)(25,29)(26,27);(6,7)(8,11)(9,10)(12,13)(14,15)(16,18)(17,19)(20,21)(22,24,26,25)(23,28,27,29);(6,7)(8,11)(9,10)(12,13);(6,8)(7,11)(9,12)(10,13)(14,16,17,20)(15,18,19,21)(22,25,26,24)(23,29,27,28)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28006);