Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12948 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1864 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
384 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)(7,13,16,15)(8,18,19,36)(9,21)(10,25)(11,24)(14,31)(17,26,28,22)(20,27,29,23)(30,37)(32,34);(6,7,11,28,12,16,24,17)(8,14,29,33,19,31,20,35)(9,22,21,13,10,26,25,15)(18,30,27,37,36,34,23,32);(6,8)(7,14)(9,23)(10,27)(11,29)(12,19)(13,30)(15,34)(16,31)(17,35)(18,21)(20,24)(22,32)(25,36)(26,37)(28,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5936);