Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#32734 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
766 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)(16,18)(17,19)(20,21)(22,23,26,28)(24,27,32,35)(25,29,33,37)(30,34,31,36);(6,8)(7,10)(9,13)(11,12)(14,16)(15,18)(17,20)(19,21)(22,24)(23,27)(25,30)(26,32)(28,35)(29,34)(31,33)(36,37);(6,8)(7,11)(9,13)(10,12)(15,19)(18,21)(24,30)(25,33)(27,34)(29,37)(31,32)(35,36);(8,13)(10,11)(16,20)(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, 25722);