Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12547 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C42 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#97 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,19,15,21,29,38,16,20,30,37,13,18,27,35)(12,22,25,39,26,17,31,33,23,34,28,41,24,32,40,36);(7,9)(10,12)(11,17,20,32)(13,24)(14,28)(15,26)(16,23)(18,22,21,34)(19,36,37,33)(25,30)(27,31)(29,40)(35,41,38,39);(12,23)(17,32)(22,34)(24,26)(25,28)(31,40)(33,36)(39,41)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5535);