Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,10)(11,13)(14,15)(16,26,31,41)(17,32)(18,22)(19,36)(20,25)(21,29,39,34)(23,27)(24,33)(28,38,43,45)(30,37,44,42)(35,40);(7,10)(11,12)(14,16,27,43,33,44,18,34,20,31,17,28,19,30,35,29)(15,21,22,37,36,45,23,41,25,39,40,42,24,38,32,26);(6,8,9,13)(7,11,10,12)(14,17,33,35,20,27,19,18)(15,22,36,23,25,40,24,32)(16,28,44,29,31,43,30,34)(21,37,45,41,39,42,38,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6276);