Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1874 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,18,11,17,28,16,14,19,27,30,13,8,10,21)(9,20,24,35,29,15,26,33,25,36,31,32,23,34,22,37);(6,8,11,19,14,17,13,7)(9,20,29,34,25,36,23,15)(10,21,12,30,28,16,27,18)(22,35,24,37,26,32,31,33);(6,9)(7,15)(8,20)(10,26)(11,29)(12,31)(13,23)(14,25)(16,35)(17,36)(18,37)(19,34)(21,32)(22,28)(24,27)(30,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5995);