Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#10679 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,17,20,19,16,18,21)(23,26)(27,28);(6,8,9,13)(7,10,11,12)(14,15)(16,19)(17,21)(18,20)(22,23,24,26)(25,27,29,28);(14,16,19,15)(17,20,18,21)(23,27)(25,29)(26,28)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3667);