Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17604 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,12,14,25)(9,20)(13,24,28,35)(15,26,16,21)(18,32)(22,34)(27,33,29,36)(30,37);(6,7)(8,13)(9,21)(10,15)(11,16)(12,22)(14,23)(17,27)(18,33)(19,29)(20,25)(24,30)(26,34)(28,31)(32,35)(36,37);(6,8,10,19)(7,13,15,29)(9,18,20,32)(11,17,23,31)(12,24,26,36)(14,28,16,27)(21,33,25,35)(22,30,34,37);(8,17)(13,27)(18,30)(19,31)(24,33)(28,29)(32,37)(35,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10592);