Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#1378 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | #336 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,6,10,7,12,17,24,8,11,18,23,19,25,30,33)(5,13,15,26,16,27,29,34,14,9,20,21,28,22,31,32);(3,5)(4,9,11,13)(6,15)(7,16)(8,14)(10,21,23,26)(12,22,25,27)(17,29)(18,20)(19,28)(24,32,33,34)(30,31)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5540);