Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#106 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11)(12,20,22,36)(13,24)(14,17)(15,29)(16,21,31,37)(18,27)(19,28)(23,38,30,34)(25,32,40,35)(26,39)(33,41);(10,12,15,30,14,22,28,23)(11,16,19,35,17,31,29,32)(13,25,27,21,26,40,41,37)(18,34,24,20,33,38,39,36);(6,7,8,9)(11,17)(16,31)(18,33)(19,29)(20,36)(24,39)(32,35)(34,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6172);