Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | H(5,𝔽2)×D25 |
| Fitting | C25×H(5,𝔽2) ⋊ C2 |
| Sylow | C25 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D25 |
| commutator, cosocle | C2×C25 | C25 |
| Frattini, socle | C2×C5 | C24×D5 |
| Fitting | C25×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | D25 |
| hypocenter | C25 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C25 ▷ C25 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
1347 conjugacy classes of subgroups; only the 513 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 513 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 200 |
238 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×#1035, D25×#453, D25×B(4,𝔽2), … |
| maximal normal subgroup of | H(5,𝔽2)×#1035, D25×#453, D25×B(4,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 2048);