Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C5 |
| Frattini, cosocle | C42 | C23×C5 |
| prime radical | C23×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19)(22,23)(24,31,33,36)(25,34)(26,29)(27,30)(28,32,37,35);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19)(22,24)(23,28)(25,32)(26,35)(27,33)(29,36)(30,37)(31,34);(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5127);