Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#10950 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1437 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17,20,13)(9,21)(10,25)(11,16)(12,15)(14,23)(18,32,33,37)(19,28,26,29)(22,35)(24,30)(27,34,31,36);(6,8,11,19,12,20,10,26)(7,13,16,29,15,17,25,28)(9,18,24,34,22,33,23,36)(14,27,21,32,30,31,35,37);(6,9,10,23,12,22,11,24)(7,14,16,21,15,30,25,35)(8,18,26,36,20,33,19,34)(13,27,29,32,17,31,28,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3938);