Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#34970 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C5×#2428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2142 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)(6,7,10,11)(8,12,9,13)(14,15,18,19)(16,20,17,21)(24,28)(26,29);(7,11)(12,13)(15,19)(20,21)(22,23)(24,29)(25,27)(26,28);(6,8,10,9)(7,12,11,13)(15,19)(20,21)(22,24,25,28)(23,26,27,29);(6,9,10,8)(7,13,11,12)(14,16,18,17)(15,20,19,21)(23,27)(26,29);(14,17,18,16)(15,21,19,20)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27958);