Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C52 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×(D5⋊AGL(1,𝔽5))⋊C2 |
| commutator, cosocle | C2×C5⋊D5 | C24 |
| Fitting | C22×C52 | #41 |
| socle | C2×C52 | C2×#41 |
| hypercenter | C22 | (D5⋊AGL(1,𝔽5))⋊C2 |
| hypocenter | C52 | #538 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
410 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^6,5^2|b^2=e,c^a=ce,d^a=df,g^a=g^4,h^a=g^4h,c^b=cf,g^b=g^2,h^b=h^2,g^c=g^4h^2,g^e=g^4,h^e=h^440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9861);