Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17764 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
964 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)(6,7,10,12,11,8,9,13)(14,15,18,24,19,23,17,22)(16,21,25,29,26,20,27,28);(6,8)(7,11)(9,13)(10,12)(14,16)(15,20)(17,25)(18,27)(19,26)(21,23)(22,28)(24,29);(6,9,11,10)(7,12,8,13)(14,16)(15,21)(17,27)(18,25)(19,26)(20,23)(22,28)(24,29);(14,17,19,18)(15,22,23,24)(16,25,26,27)(20,28,21,29)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10752);