Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
630 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)(6,7,10,16,32,39,62,65)(8,18,21,47,20,53,30,61)(9,24,27,56,26,42,58,67)(11,14,31,38,12,17,33,45)(13,34,37,19,49,51,44,50)(15,40,43,29,46,60,69,28)(22,48,54,68,23,41,55,66)(25,35,57,63,52,36,59,64);(6,8,11,22)(7,13,14,35)(9,25,28,50)(10,30,31,55)(12,23,32,20)(15,41,24,53)(16,44,38,64)(17,36,39,49)(18,46,48,42)(19,26,52,29)(21,33,54,62)(27,59,40,51)(34,58,57,60)(37,45,63,65)(43,68,56,47)(61,69,66,67);(7,14)(9,26)(10,31)(13,35)(15,42)(17,39)(18,41)(19,50)(21,54)(24,46)(25,52)(27,60)(28,29)(30,55)(33,62)(34,59)(36,49)(40,58)(43,69)(47,61)(48,53)(51,57)(56,67)(66,68)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1478);