Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33628 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15)(16,24)(17,28)(18,33)(19,21)(20,25)(22,31)(23,30)(26,44)(27,41)(29,39)(32,40)(34,38)(35,43)(36,45)(37,42);(6,8,9,13)(7,10,11,12);(14,16,19,27)(15,20,21,35)(17,29,32,45)(18,34,30,26)(22,38,37,44)(23,39,33,36)(24,40,41,28)(25,42,43,31);(15,21)(20,35)(22,37)(23,33)(24,41)(28,40)(36,39)(38,44)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26616);