Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C5 |
No aliases for this group.
| Fitting | C52×C4⋊Q8 ⋊ C2 |
| Sylow | C52 ⋊ #532 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C52×C4⋊Q8 | C2 |
| hypercenter | C4⋊Q8 | C5⋊D5 |
| hypocenter | C52 | #532 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
1240 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 267 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 800 |
Polycyclic presentation:
2^6,5^2|b^2=f,c^2=f,d^2=f,c^a=cf,d^a=df,g^a=g^4,h^a=h^4,c^b=cf,d^b=df,d^c=de196 conjugacy classes — view the full character table.
| maximal subgroup of | #72220969, C5⋊D5×#532, #72221191, … |
| maximal normal subgroup of | #72220969, C5⋊D5×#532, #72221191, … |
| maximal quotient of | #72223091 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9606);