Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C25×2−1+4 ⋊ C2 |
| Sylow | C25 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D25 |
| commutator, cosocle | C2×C25 | C25 |
| Frattini, socle | C2×C5 | C24×D5 |
| Fitting | C25×2−1+4 | C2 |
| hypercenter | 2−1+4 | D25 |
| hypocenter | C25 | #585 |
| LCS | G ▷ C2×C25 ▷ C25 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
1215 conjugacy classes of subgroups; only the 511 normal subgroups are shown.
This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 511 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^6,5^2|c^2=f,e^2=f,g^5=h,e^a=ef,g^a=g^4h^4,h^a=h^4,d^b=df,e^c=ef238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 2051);