Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C52 ⋊ #322 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D5×AGL(1,𝔽5) |
| commutator | C2×C52 | C4×C8 |
| Fitting | C2×C4×C52 | C2×C4 |
| socle, prime radical | C22×C52 | C42 |
| cosocle | C4×C5×Dic5 | C22 |
| hypocenter | C52 | #322 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2×C4 | |
143 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 66 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=c,c^2=e,d^2=f,b^a=bf,h^a=h^4,g^b=g^2,g^c=g^4136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 2099);