Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10771 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #381 |
| 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 | |
1164 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)(8,12,17,25)(9,19,20,13)(10,22)(11,14)(15,23)(16,28)(18,32,31,26)(21,27,34,33)(24,29)(30,35)(36,37);(6,8,10,18,11,17,23,31)(7,12,15,26,14,25,22,32)(9,16,21,30,20,29,34,37)(13,24,27,35,19,28,33,36);(6,9)(7,13)(8,16)(10,21)(11,20)(12,24)(14,19)(15,27)(17,29)(18,30)(22,33)(23,34)(25,28)(26,35)(31,37)(32,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3759);