Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
306 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)(7,11)(8,9)(12,13)(14,15,18,24,20,25,33,23,19,22,34,39,32,40,17,31)(16,26,28,36,30,41,44,43,29,42,45,21,35,37,27,38);(6,7)(8,11)(9,13)(10,12)(14,16,19,29)(15,21,22,36)(17,28,33,45)(18,27,34,44)(20,35,32,30)(23,41,31,37)(24,42,39,26)(25,43,40,38);(6,8,10,9)(7,11,12,13)(15,22)(21,36)(23,31)(24,39)(25,40)(26,42)(37,41)(38,43)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4972);