Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7869 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C22×C42×C5 ◁ G | |
684 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)(8,10)(9,12)(11,13)(14,15,17,20,18,21,24,27)(16,19,23,26,22,25,28,29);(7,10,12,11)(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29);(16,22)(19,25)(23,28)(26,29)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 857);