Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#9310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15)(16,20,25,36)(17,27)(18,31)(19,22)(21,29)(23,33)(24,40,28,45)(26,43,42,35)(30,39)(32,38)(34,44,37,41);(6,7,8,13,10,12,9,11)(14,16,19,25)(15,20)(17,28,30,24)(18,26,33,42)(21,37)(22,36)(23,35)(27,45)(29,44,32,41)(31,43)(34,38)(39,40);(6,8,10,9)(7,11,12,13)(14,17,19,30)(15,21,22,38)(16,24,25,28)(18,32,33,29)(20,34,36,37)(23,39,31,27)(26,44,42,41)(35,45,43,40)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2298);