Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15,20,25)(16,21,28,37)(17,29,31,40)(18,24,19,23)(22,33,39,32)(26,38,42,45)(27,41,34,36)(30,35,43,44);(6,7,10,12)(8,11,9,13)(14,16)(15,21)(17,30)(18,27)(19,34)(20,28)(22,38)(23,36)(24,41)(25,37)(26,33)(29,44)(31,43)(32,42)(35,40)(39,45);(6,8,10,9)(7,11,12,13)(14,17)(15,22)(16,26)(18,33)(19,32)(20,31)(21,35)(23,40)(24,29)(25,39)(27,43)(28,42)(30,34)(36,45)(37,44)(38,41)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3175);