Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1864 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
384 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)(7,12,15,14)(8,17,18,36)(9,20)(10,24)(13,28)(16,33,25,21)(19,22,26,37)(23,35)(27,29)(30,32);(6,7,10,25,11,15,24,16)(8,13,26,31,18,28,19,34)(9,21,20,12,23,33,35,14)(17,32,22,30,36,27,37,29);(6,8)(7,13)(9,22)(10,26)(11,18)(12,27)(14,32)(15,28)(16,34)(17,35)(19,24)(20,36)(21,30)(23,37)(25,31)(29,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5940);