Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1336 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)(9,20)(10,14)(11,15)(13,22)(17,34,35,27)(18,32,25,28)(21,37,36,29)(23,33)(24,31)(26,30);(6,8)(7,12)(9,21)(10,19)(11,25)(13,29)(14,16)(15,32)(17,23)(18,24)(20,37)(22,36)(26,35)(27,30)(28,31)(33,34);(6,9)(7,13)(8,17)(10,22)(11,26)(12,27)(14,20)(15,33)(16,34)(18,36)(19,35)(21,25)(23,24)(28,37)(29,32)(30,31);(6,8)(7,12)(9,17)(10,19)(11,18)(13,27)(14,16)(15,28)(20,34)(21,23)(22,35)(24,25)(26,36)(29,30)(31,32)(33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17397);