Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8132 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | C23 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ G | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,16,19,17,18,20,21);(7,10)(11,13)(22,23,24,26,25,27,28,29);(6,8,9,12)(7,11,10,13)(15,18)(19,21)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1120);