Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13236 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11,12,13)(14,15)(16,27)(17,33)(18,36)(19,37)(20,26)(21,29)(22,30)(23,35)(24,28)(25,34)(31,45)(32,38)(39,42)(40,44)(41,43);(6,7,8,9)(11,13)(14,16,28,42,19,31,17,29,20,32,35,44,34,43,18,30)(15,21,36,45,25,39,23,27,26,22,33,41,37,40,24,38);(14,16,28,42,19,31,17,29,20,32,35,44,34,43,18,30)(15,22,36,41,25,40,23,38,26,21,33,45,37,39,24,27)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6224);