Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7318 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#7318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1304 |
| commutator | C22×C4 | C42×C5 |
| Frattini, cosocle | C22×#41 | C22×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×#41 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,10,14)(8,16,18,30)(9,13,20,27)(11,15,22,29)(12,23,24,17)(19,26,32,36)(21,28,33,37)(25,34,35,31);(6,8)(7,12)(9,17)(10,18,22,32)(11,19)(13,26)(14,24,29,35)(15,25)(16,28)(20,23,33,34)(21,31)(27,36,37,30)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 306);