Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10744 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | C22⋊#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1152 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)(6,7,10,16,12,17,11,15)(8,18,20,13,21,28,22,27)(9,14,24,29,25,30,23,31)(19,33,36,37,35,32,34,26);(6,8,11,22,12,21,10,20)(7,13,16,28,17,27,15,18)(9,19,23,34,25,35,24,36)(14,26,29,33,30,37,31,32);(6,9)(7,14)(8,19)(10,24)(11,23)(12,25)(13,26)(15,31)(16,29)(17,30)(18,32)(20,36)(21,35)(22,34)(27,37)(28,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3732);