Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
494 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)(6,7,10,13,11,15,9,14)(8,16,18,20,19,21,17,12)(22,23)(24,32,25,28)(26,31)(27,30)(29,34,36,33)(35,37);(6,8)(7,12,15,20)(9,17)(10,18)(11,19)(13,16,14,21)(22,24,26,25)(23,28)(27,34,35,33)(29,30)(31,32)(36,37);(7,13)(9,10)(12,16)(14,15)(17,18)(20,21)(22,25,26,24)(23,29,31,36)(27,33,35,34)(28,30,32,37)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2165);