Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ #338 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | AGL(1,𝔽5)2 |
| commutator, Fitting | C4×C52 | C42 |
| socle | C2×C52 | #95 |
| prime radical | C2×D52 | Q8 |
| cosocle | C4×D52 | C22 |
| hypocenter | C52 | #338 |
| LCS | G ▷ C4×C52 ▷ C52 | |
| UCS | C1 ◁ C4 | |
183 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 44 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|a^2=c,b^2=d,c^2=f,d^2=f,e^2=f,b^a=be,d^a=df,h^a=h^2,c^b=cf,g^b=g^2,h^c=h^4,g^d=g^458 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5608);