Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11492 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
360 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)(14,15,19,25,21,24,34,42,20,26,33,43,18,35,17,23)(16,27,30,41,29,22,28,38,31,40,44,37,36,39,32,45);(7,10)(8,12)(14,16)(15,22)(17,32)(18,36)(19,30)(20,31)(21,29)(23,41)(24,40)(25,38)(26,39)(27,35)(28,34)(33,44)(37,42)(43,45);(14,17,18,33,20,34,21,19)(15,23,35,43,26,42,24,25)(16,28,36,30,31,32,29,44)(22,37,27,38,39,41,40,45)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4480);