Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #99 |
| commutator | C22×C4 | C2×C4×C5 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C5 |
| prime radical | C23×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C2×C5×C4⋊C4 ◁ G | |
116 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(7,10,11,12)(14,15,17,21,18,20,24,27)(16,19,23,25,22,26,29,28);(6,7)(8,12)(9,11)(10,13)(14,16)(15,19,20,26)(17,23)(18,22)(21,28,27,25)(24,29)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 486);