Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ #336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | AGL(1,𝔽5)2 |
| commutator | C2×C52 | C4×C8 |
| Fitting, socle | C22×C52 | C42 |
| prime radical | C22×C5⋊D5 | C2×C4 |
| cosocle | C2×#6036 | C22 |
| hypocenter | C52 | #336 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 | |
263 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 64 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=c,b^2=d,c^2=f,d^2=f,b^a=be,h^a=h^2,g^b=g^2,h^c=h^4,g^d=g^476 conjugacy classes — view the full character table.
| maximal subgroup of | #72221645, #72221684, #72221685 |
| maximal normal subgroup of | #72221645, #72221684, #72221685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5635);