Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#21232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(7,9)(10,11,12,13)(14,15)(16,26)(17,31)(18,23)(19,24)(20,25)(21,29)(22,33)(27,35)(28,37)(30,36)(32,39)(34,38)(40,41)(42,44)(43,45);(6,7,8,9)(11,13)(14,16,18,29,20,30,19,28)(15,21,24,26,25,37,23,36)(17,27,32,42,34,43,33,41)(22,35,38,40,39,45,31,44);(14,17,18,32,20,34,19,33)(15,22,24,38,25,39,23,31)(16,27,29,42,30,43,28,41)(21,35,26,40,37,45,36,44);(14,18,20,19)(15,23,25,24)(16,28,30,29)(17,32,34,33)(21,26,37,36)(22,31,39,38)(27,41,43,42)(35,40,45,44)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14220);