Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11717399 |
| commutator, cosocle | C4×C5⋊D5 | C23 |
| Frattini | C4 | C2×D52⋊C2 |
| Fitting | C52×D4 | D4 |
| socle | C2×C52 | #120 |
| hypercenter | D4 | D52⋊C2 |
| hypocenter | C52 | #454 |
| LCS | G ▷ C4×C5⋊D5 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
275 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 31 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,e^2=f,b^a=bd,c^a=cf,d^a=df,g^a=g^4,h^a=g^2h,c^b=ce,e^b=ef,g^b=g^3h^3,h^b=h^2,e^c=ef,g^d=g^4,h^d=h^455 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 6683);