Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1301 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22⋊C4 |
| commutator | C2×C4 | C42×C5 |
| Frattini, cosocle | C2×C42 | C22×C5 |
| socle | C22×C5 | #95 |
| prime radical | C23×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
148 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 320 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15,17,20,18,21,25,28)(16,22,24,29,23,19,26,27);(6,7,9,12,10,13,8,11)(14,16)(15,19,21,22)(17,24)(18,23)(20,27,28,29)(25,26)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 434);