Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13233 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1964 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,12,17,13,18,11,33)(8,19,23,46,24,47,34,45)(9,25,28,42,29,56,27,15)(10,30,20,43,32,16,35,37)(14,36,39,31,40,53,63,22)(21,49,51,44,52,66,60,67)(26,57,48,62,59,55,61,41)(38,50,54,64,65,58,69,68);(6,8,20,36)(7,14,37,46)(9,21,48,64)(10,31,12,34)(11,23,32,22)(13,24,35,53)(15,38,57,66)(16,19,17,39)(18,40,43,45)(25,54,41,44)(26,58,28,60)(27,51,59,50)(29,52,61,68)(30,47,33,63)(42,65,55,49)(56,69,62,67);(6,9,12,28,13,29,11,27)(7,15,33,56,18,42,17,25)(8,21,34,60,24,52,23,51)(10,26,35,61,32,59,20,48)(14,38,63,69,40,65,39,54)(16,41,37,57,30,62,43,55)(19,44,46,66,47,67,45,49)(22,50,36,64,31,58,53,68)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6221);