Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13243 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1970 |
| prime radical | C2×C5×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
410 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)(8,20,25,44)(9,27)(10,17)(11,34)(12,36)(13,19)(14,23,39,32)(15,29)(16,21)(18,35)(22,37,52,63)(24,40,48,62)(26,38,54,47)(28,56)(30,41)(31,58)(33,45)(42,60)(43,49)(46,66,65,51)(50,67,59,55)(53,69,68,64)(57,61);(6,8,21,48,35,54,10,32,13,25,11,24,12,26,33,23)(7,14,17,38,36,62,16,44,19,39,45,47,18,40,34,20)(9,22,49,66,61,68,28,59,31,52,29,51,30,53,60,50)(15,37,27,55,56,69,41,65,43,63,58,67,42,64,57,46);(6,9,10,28,12,30,21,49,13,31,33,60,35,61,11,29)(7,15,34,57,18,42,45,58,19,43,16,41,36,56,17,27)(8,22,32,59,26,53,48,66,25,52,23,50,54,68,24,51)(14,37,20,46,40,64,47,67,39,63,44,65,62,69,38,55)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6231);