Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10756 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | C22⋊#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,11,14,22,28)(8,12,18,25,17,24,21,29)(9,13,20,26,19,27,33,37)(16,30,32,36,31,23,34,35);(6,8)(7,12,14,24)(9,16)(10,21)(11,17)(13,23,27,30)(15,29,28,25)(18,22)(19,31)(20,34)(26,36,37,35)(32,33);(6,9)(7,13)(8,16)(10,20)(11,19)(12,23)(14,27)(15,26)(17,31)(18,32)(21,34)(22,33)(24,30)(25,35)(28,37)(29,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3744);