Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#66285 |
| Fitting | C52×H(5,𝔽2) ⋊ C2 |
| Sylow | C52 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C24×D5 |
| commutator, cosocle | C2×C5 | C25×C5 |
| Frattini | C2 | C24×C5×D5 |
| Fitting | C52×H(5,𝔽2) | C2 |
| socle | C2×C52 | C25 |
| hypercenter | C5×H(5,𝔽2) | D5 |
| hypocenter | C5 | C5×#585 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C5 ◁ C5×H(5,𝔽2) | |
1786 conjugacy classes of subgroups; only the 886 normal subgroups are shown.
This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 886 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
340 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72221786 |
| maximal normal subgroup of | #72221786 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 10222);