Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11090 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #393 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1439 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
852 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)(6,7,10,15,11,14,24,29)(8,12,18,27,17,26,33,36)(9,19,22,30,21,13,23,28)(16,31,32,37,20,25,34,35);(6,8)(7,12)(9,20,21,16)(10,18)(11,17)(13,25,19,31)(14,26)(15,27)(22,34,23,32)(24,33)(28,35,30,37)(29,36);(6,9)(7,13)(8,16)(10,23)(11,21)(12,25)(14,19)(15,30)(17,20)(18,34)(22,24)(26,31)(27,37)(28,29)(32,33)(35,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4078);