Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#11597 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
394 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)(6,7,8,10,9,11,12,13)(14,15)(16,25,28,42)(17,21)(18,32)(19,24)(20,27,36,44)(22,30)(23,31)(26,41,29,38)(33,40,45,35)(34,37)(39,43);(14,16)(15,20)(17,26)(18,33)(19,28)(21,35)(22,38)(23,40)(24,36)(25,37)(27,43)(29,31)(30,44)(32,42)(34,45)(39,41);(14,17,19,31)(15,21,24,23)(16,26,28,29)(18,30,34,43)(20,35,36,40)(22,37,39,32)(25,41,42,38)(27,33,44,45)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4585);