Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#1540 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C23×C5 | C2×C22⋊C4 |
| prime radical | C2×C5×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4,6,8,7,10,5,9)(11,12,15,20,16,19,14,18)(13,21,22,26,23,17,24,25);(4,8)(5,6)(9,10)(11,13)(12,17)(14,22)(15,24)(16,23)(18,25)(19,21)(20,26);(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10683);