Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13192 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| 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 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,19,15,18,28,31,16,20,29,34,13,27,22,37)(12,21,25,36,26,32,39,40,23,35,38,41,24,17,30,33);(10,12,22,30,13,24,29,38,16,23,28,39,15,26,14,25)(11,17,31,36,27,35,19,33,20,32,37,41,18,21,34,40);(6,7,8,9)(12,23)(17,32)(21,35)(24,26)(25,38)(30,39)(33,40)(36,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6180);