Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11319 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #401 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1443 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C24 ▷ 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)(6,7,10,15,11,14,23,30)(8,12,18,26,17,25,22,31)(9,19,21,34,20,13,27,29)(16,24,33,36,32,28,35,37);(6,8)(7,12,14,25)(9,16,20,32)(10,22)(11,17)(13,28)(15,31,30,26)(18,23)(19,24)(21,35,27,33)(29,36)(34,37);(6,9)(7,13,14,19)(8,16,17,32)(10,21)(11,20)(12,24)(15,29,30,34)(18,33,22,35)(23,27)(25,28)(26,36)(31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4307);