Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8010 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#95 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C42×C5 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,11,15,10,14)(8,16,18,12,19,20,17,21)(22,23)(24,30,32,27)(25,28)(26,29)(31,36,33,35)(34,37);(6,8,10,17,11,19,9,18)(7,12,13,20,15,21,14,16)(22,24)(23,27)(25,33)(26,32)(28,36)(29,30)(31,34)(35,37);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,16,21,20)(22,25,26,34)(23,28,29,37)(24,31,32,33)(27,35,30,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 998);