Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11302 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #397 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1443 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
532 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)(6,7,11,17,12,15,10,16)(8,13,21,30,19,29,20,28)(9,22,24,31,25,32,26,14)(18,33,34,37,23,36,35,27);(6,8,10,21,12,19,11,20)(7,13,17,28,15,29,16,30)(9,23,24,34,25,18,26,35)(14,27,32,33,31,37,22,36);(6,9)(7,14,15,31)(8,18,19,23)(10,24)(11,26)(12,25)(13,27)(16,22,17,32)(20,34,21,35)(28,33)(29,37)(30,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4290);