Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#11610 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C23×C5 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
326 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)(7,10)(8,12)(11,13)(14,15,17,22,19,23,30,37,18,21,29,34,31,35,42,44)(16,24,26,36,27,38,40,45,25,20,28,32,39,33,41,43);(6,7,9,11)(8,13,12,10)(14,16,18,25)(15,20,21,24)(17,28,29,26)(19,27,31,39)(22,36,34,32)(23,33,35,38)(30,41,42,40)(37,45,44,43);(16,25)(20,24)(26,28)(27,39)(32,36)(33,38)(40,41)(43,45)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4598);