Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#32865 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,12)(8,11,13,10)(14,15)(16,19)(17,18)(20,21)(22,23)(24,27)(25,29)(26,28)(30,34)(31,36)(32,35)(33,37);(6,8)(7,10)(9,13)(11,12)(15,18)(19,21)(22,24,26,32)(23,27,28,35)(25,30,33,31)(29,34,37,36);(6,8)(7,11)(9,13)(10,12)(15,18)(16,20)(24,30)(25,33)(27,34)(29,37)(31,32)(35,36);(8,13)(10,11)(14,16,17,20)(15,19,18,21)(23,28)(27,35)(29,37)(34,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25853);