Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#29548 |
| 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 | |
698 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)(8,12)(9,11)(10,13)(22,23,26,30)(24,27,31,35)(25,28,33,29)(32,36,37,34);(6,8,9,13)(7,10,11,12)(14,15,17,18)(16,19,20,21)(22,24)(23,27)(25,32)(26,31)(28,36)(29,34)(30,35)(33,37);(15,18)(19,21)(23,28)(24,31)(25,33)(27,34)(29,30)(35,36);(7,11)(10,12)(14,16,17,20)(15,19,18,21)(24,31)(27,35)(32,37)(34,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22536);