Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10743 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | C22⋊#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1208 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)(6,7,11,15,12,16,10,24)(8,17,21,27,20,32,19,13)(9,14,22,29,23,30,25,28)(18,33,34,26,35,31,36,37);(6,8)(7,13,16,27)(9,18)(10,19)(11,21)(12,20)(14,26,30,37)(15,17,24,32)(22,34)(23,35)(25,36)(28,33,29,31);(6,9)(7,14)(8,18)(10,25)(11,22)(12,23)(13,26)(15,29)(16,30)(17,31)(19,36)(20,35)(21,34)(24,28)(27,37)(32,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3731);