Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12720 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #410 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | #1782 |
| prime radical | C5×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C42×C5 ◁ C4×C5×#41 ◁ G | |
694 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)(8,17)(9,22,24,14)(10,15)(11,16)(12,19,28,18)(13,23,30,37)(20,29)(21,33)(25,31)(26,36)(27,35,32,34);(6,8)(7,12)(9,19)(10,20)(11,21)(13,29)(14,17)(15,27)(16,28)(18,24)(22,33)(23,35)(25,36)(26,30)(31,32)(34,37);(6,9)(7,13,16,30)(8,18)(10,23)(11,24)(12,26,28,29)(14,15,22,31)(17,32,33,27)(19,21)(20,34)(25,37)(35,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5708);