Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C52 ⋊ #386 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D52 |
| commutator | C22×C52 | C22×C4 |
| Fitting | C24×C52 | C22 |
| socle, cosocle | C23×C52 | C23 |
| prime radical | C22×C5×C22⋊D5 | C2 |
| hypercenter | C24 | D52 |
| hypocenter | C52 | #386 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C23 ◁ C24 | |
808 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 808 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^6,5^2|b^2=e,b^a=bf,c^a=cd,g^a=g^4,c^b=cd,h^b=h^4172 conjugacy classes — view the full character table.
| maximal subgroup of | #72223419 |
| maximal normal subgroup of | #72223419 |
| maximal quotient of | #72223415, #72226411 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 4241);