Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8486 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3559 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,11,13,16)(12,20,21,19)(22,23,26,30,27,31,25,35)(24,32,33,28,34,29,36,37);(6,7)(8,12)(9,16)(10,13)(11,18)(14,19)(15,21)(17,20)(22,24)(23,28)(25,36)(26,33)(27,34)(29,30)(31,37)(32,35);(6,8,10,15)(7,12,13,21)(9,17,18,14)(11,19,16,20)(22,24)(23,29)(25,33)(26,36)(27,34)(28,30)(31,32)(35,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1474);