Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C5×#3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C24×C5 ◁ G | |
1318 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)(8,16,19,12)(9,13)(10,22,23,14)(11,15)(17,24,31,26)(18,25)(20,33,34,27)(21,28)(29,32)(30,36)(35,37);(6,8)(7,12)(9,17)(10,18)(11,19)(13,26)(14,25)(15,16)(20,30)(21,31)(22,29)(23,32)(24,28)(27,36)(33,35)(34,37);(6,9,11,21)(7,13,15,28)(8,17,19,31)(10,20,23,34)(12,24,16,26)(14,27,22,33)(18,30,32,37)(25,35,29,36);(6,10)(7,14)(8,18)(9,20)(11,23)(12,25)(13,27)(15,22)(16,29)(17,30)(19,32)(21,34)(24,35)(26,36)(28,33)(31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14765);