Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
436 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)(16,20,25,35)(17,27)(18,30)(19,22)(21,28)(23,32)(24,38,39,45)(26,42,41,34)(29,37)(31,36)(33,43,44,40);(6,7,9,11,10,13,8,12)(14,16)(15,20,22,35)(17,24)(18,26)(19,25)(21,33,36,44)(23,34,30,42)(27,38,37,45)(28,40)(29,39)(31,43)(32,41);(6,8,10,9)(7,12,13,11)(14,17,19,29)(15,21,22,36)(16,24,25,39)(18,31,32,28)(20,33,35,44)(23,37,30,27)(26,43,41,40)(34,45,42,38)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2292);