Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21014 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1748 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)(6,7)(8,12,18,25)(9,19,21,13)(10,14)(11,15)(16,23)(17,24,30,35)(20,32,33,26)(22,27)(28,34)(29,31)(36,37);(6,8)(7,12,15,25)(9,16)(10,17)(11,18)(13,23,19,31)(14,24,27,35)(20,28)(21,29)(22,30)(26,34,32,37)(33,36);(6,9)(7,13)(8,16)(10,20)(11,21)(12,23)(14,26)(15,19)(17,28)(18,29)(22,33)(24,34)(25,31)(27,32)(30,36)(35,37);(6,10,11,22)(7,14,15,27)(8,17,18,30)(9,20,21,33)(12,24,25,35)(13,26,19,32)(16,28,29,36)(23,34,31,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14002);