Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23525 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#394 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,18)(9,23)(10,15)(11,16)(12,17)(13,21)(14,26)(19,30)(20,29)(22,33)(24,28)(25,32)(27,31)(34,36)(35,37);(6,8)(7,13)(9,24)(10,20)(11,21)(12,22)(14,30)(15,18)(16,33)(17,29)(19,27)(23,37)(25,36)(26,35)(28,32)(31,34);(6,9,10,25,12,27,11,26)(7,14,16,31,17,32,15,23)(8,19,21,36,22,24,20,35)(13,28,18,34,29,30,33,37);(6,10,12,11)(7,15,17,16)(8,20,22,21)(9,25,27,26)(13,18,29,33)(14,23,32,31)(19,35,24,36)(28,34,30,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16513);