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×#34921 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C5×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2014 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)(14,15,17,18)(16,19,20,21)(22,23,25,27)(24,28,29,26);(7,11)(12,13)(15,18)(19,21)(22,24)(23,26)(25,29)(27,28);(15,18)(19,21)(23,27)(24,29);(6,8,10,9)(7,12,11,13)(14,16,17,20)(15,19,18,21)(24,29)(26,28);(6,9,10,8)(7,13,11,12)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27909);