Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12953 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1864 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,16,12,17,11,29)(8,18,21,33,23,30,22,37)(9,24,26,34,28,14,27,15)(13,25,31,19,32,36,35,20);(6,8)(7,13,17,32)(9,25)(10,22)(11,21)(12,23)(14,30,24,18)(15,33,34,37)(16,35,29,31)(19,27)(20,26)(28,36);(7,14)(9,26)(10,11)(13,30)(15,29)(16,34)(17,24)(18,32)(19,36)(20,25)(21,22)(27,28)(31,33)(35,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5941);