Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,9,13)(8,16,17,20,19,21,18,12)(22,23,26,30,27,31,25,29)(24,32,34,28,35,36,33,37);(6,8)(7,12,15,20)(9,18)(10,17)(11,19)(13,21,14,16)(22,24)(23,28)(25,34)(26,33)(27,35)(29,36)(30,32)(31,37);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,16,20,21)(22,24)(23,28)(25,33)(26,34)(27,35)(29,32)(30,36)(31,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2316);