Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23992 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1128 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)(6,7,11,15)(8,12,18,26)(9,13,19,27)(10,21,23,14)(16,29,22,24)(17,31,32,25)(20,34,33,28)(30,36,35,37);(8,16,18,22)(12,24,26,29)(17,30,32,35)(25,36,31,37);(6,8)(7,12)(9,17)(10,22)(11,18)(13,25)(14,29)(15,26)(16,23)(19,32)(20,35)(21,24)(27,31)(28,37)(30,33)(34,36);(6,9)(7,13)(8,17)(10,20)(11,19)(12,25)(14,28)(15,27)(16,30)(18,32)(21,34)(22,35)(23,33)(24,36)(26,31)(29,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16980);