Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12944 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1862 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
388 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)(6,7,11,17,13,18,12,31)(8,19,23,35,24,15,28,34)(9,25,22,33,26,32,21,14)(10,27,20,36,30,37,29,16);(6,8,11,23,13,24,12,28)(7,14,17,25,18,33,31,32)(9,20,22,30,26,29,21,10)(15,16,34,27,19,36,35,37);(6,9)(7,15)(8,20)(10,28)(11,22)(12,21)(13,26)(14,16)(17,34)(18,19)(23,30)(24,29)(25,27)(31,35)(32,37)(33,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5932);