Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#34604 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,17,18)(16,19,20,21)(22,23)(24,27)(25,28)(26,29)(30,34)(31,35)(32,36)(33,37);(7,10)(11,13)(22,24,26,31)(23,27,29,35)(25,32,33,30)(28,36,37,34);(15,18)(19,21)(22,25,26,33)(23,28,29,37)(24,30,31,32)(27,34,35,36);(14,16,17,20)(15,19,18,21)(23,29)(27,35)(28,37)(34,36);(6,8,9,12)(7,11,10,13)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27592);