Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,10,12,16,19)(8,11,15,18,14,17,20,21)(22,23,27,30,28,31,26,35)(24,32,33,36,25,29,34,37);(6,8)(7,11)(9,15)(10,14)(12,17)(13,18)(16,20)(19,21)(22,24)(23,29)(25,28)(26,33)(27,34)(30,36)(31,32)(35,37);(8,14)(11,17)(15,20)(18,21)(22,25,28,24)(23,29,31,32)(26,33,27,34)(30,37,35,36);(7,12)(11,17)(13,19)(18,21)(22,26,28,27)(23,30,31,35)(24,33,25,34)(29,36,32,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10397);