Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C52 ⋊ Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #6038 |
| commutator | C4×C52 | C2×C8 |
| Frattini, hypercenter | C42 | D52 |
| Fitting, cosocle | C42×C52 | C22 |
| socle, prime radical | C22×C52 | C22⋊C4 |
| hypocenter | C52 | Q8⋊C8 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
210 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^6,5^2|a^2=d,b^2=cdf,c^2=e,d^2=f,b^a=bc,c^a=ce,h^a=h^4,g^b=g^4196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1607);