Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11924 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C5×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,8,11,9,10,12,13)(14,15)(16,26,29,42)(17,22)(18,32)(19,24)(20,25)(21,28,37,45)(23,34)(27,39,30,43)(31,41)(33,36,44,38)(35,40);(7,10)(8,12)(14,16)(15,21)(17,27)(18,33)(19,30)(20,29)(22,36)(23,39)(24,38)(25,37)(26,40)(28,31)(32,42)(34,45)(35,44)(41,43);(14,17,20,19)(15,22,25,24)(16,27,29,30)(18,31,35,34)(21,36,37,38)(23,32,41,40)(26,39,42,43)(28,44,45,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4912);