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×#9170 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,15)(8,11,14,10)(9,16)(13,17)(18,20)(19,21)(22,23,26,30,27,31,25,29)(24,32,33,28,35,36,34,37);(6,8,10,7,12,14,11,15)(9,16,18,13,19,21,17,20)(22,24)(23,28)(25,33)(26,34)(27,35)(29,36)(30,32)(31,37);(6,9)(7,13)(8,16)(10,18)(11,17)(12,19)(14,21)(15,20)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2158);