Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#21996 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15)(16,19)(17,18)(20,21)(22,23,24,26,25,27,28,29);(6,8)(7,10)(9,13)(11,12);(8,13)(10,12)(15,18)(19,21);(7,11)(10,12)(14,16,17,20)(15,19,18,21)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14984);