Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#7660 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
426 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)(7,10)(8,12)(11,13)(15,19,20,21)(22,23,25,27,26,29,24,28);(6,7,9,11)(8,13,12,10)(14,15)(16,19)(17,21)(18,20)(23,27)(24,25)(28,29);(14,16,18,17)(15,19,20,21)(22,24,26,25)(23,28,29,27)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 648);