Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#8998 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C5×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
414 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)(8,15,19,21)(9,18,20,12)(10,13)(11,14)(16,17)(22,23)(24,30,32,37)(25,27)(26,29)(28,31,35,34)(33,36);(6,8,16,18,11,19,10,12)(7,9,17,15,14,20,13,21)(23,27)(26,33)(28,35)(31,34);(6,9)(7,12)(8,17)(10,21)(11,20)(13,19)(14,18)(15,16)(22,24)(23,28)(25,32)(26,34)(27,35)(29,37)(30,36)(31,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1986);