Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15)(16,18)(17,20)(19,21)(22,23)(24,27)(25,28)(26,29)(30,35)(31,34)(32,36)(33,37);(6,8,9,13)(7,10,11,12)(22,24)(23,27)(25,30)(26,31)(28,35)(29,34)(32,37)(33,36);(15,18)(16,21)(19,20)(22,25,26,32)(23,28,29,36)(24,30,31,37)(27,33,34,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3330);