Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12792 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #121 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1797 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
440 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)(6,7,11,18,12,16,23,31,13,19,22,14,10,17,8,20)(9,24,27,36,26,33,21,35,28,34,29,37,25,15,32,30);(6,8,10,22,13,23,12,11)(7,14,16,20,19,18,17,31)(9,21,25,27,28,32,26,29)(15,30,24,36,33,35,34,37);(6,9)(7,15)(8,21)(10,25)(11,29)(12,26)(13,28)(14,30)(16,24)(17,34)(18,35)(19,33)(20,36)(22,27)(23,32)(31,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5780);