Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C52 ⋊ #447 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D52⋊C2 |
| commutator, cosocle | C2×#1039 | C23 |
| Fitting | C23×C52 | D4 |
| socle | C22×C52 | C2×D4 |
| hypercenter | C23 | D52⋊C2 |
| hypocenter | C52 | #447 |
| LCS | G ▷ C2×#1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C23 | |
388 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^6,5^2|b^2=df,d^2=f,b^a=bd,d^a=df,g^a=g^4,h^a=g^2h,c^b=ce,g^b=g^3h^3,h^b=h^2,g^d=g^4,h^d=h^479 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 6731);