Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#1042 |
| commutator, socle | C22×C52 | C22×C4 |
| Fitting, prime radical | C24×C52 | C4 |
| cosocle | C2×C5×Dic5 | C23 |
| hypercenter | C24 | #1042 |
| hypocenter | C52 | #407 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C24 | |
447 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 109 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|a^2=d,d^2=f,b^a=bf,c^a=ce,g^a=g^2,h^a=h^4,g^d=g^4172 conjugacy classes — view the full character table.
| maximal subgroup of | #72221644, #72221692, #72221694 |
| maximal normal subgroup of | #72221644, #72221692, #72221694 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 7036);