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) = C42 |
| direct product | C5×#12594 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C42 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#97 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,18,24,19,21,26,28,20,25,27,29,17,23,16,22);(7,10)(11,13)(15,21,25,23)(16,26)(18,27)(22,29,28,24);(6,8,9,12)(7,11,10,13)(14,16,17,27,20,26,19,18)(15,22,23,29,25,28,21,24)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5582);