Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1936 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
296 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)(6,7,12,19,14,20,13,42)(8,21,25,44,26,53,40,54)(9,27,31,49,32,55,41,63)(10,33,11,37,36,18,35,17)(15,24,45,23,47,38,46,34)(16,29,50,30,52,64,51,39)(22,57,60,68,61,43,28,48)(56,65,66,59,69,58,67,62);(6,8,10,34,13,25,35,24,14,26,36,23,12,40,11,38)(7,15,37,21,19,46,18,54,20,47,17,53,42,45,33,44)(9,28,64,59,31,61,39,65,32,60,29,62,41,22,30,58)(16,48,49,67,51,57,27,56,52,68,63,66,50,43,55,69);(6,9,10,29,13,31,35,30,14,32,36,64,12,41,11,39)(7,16,17,49,19,51,33,27,20,52,37,63,42,50,18,55)(8,22,23,58,25,28,38,59,26,61,34,65,40,60,24,62)(15,43,21,56,46,48,54,66,47,57,53,69,45,68,44,67)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6145);