Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C83 · Schur multiplier (H2) = C25 |
| direct product | C2×#137×C83 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C83 |
| prime radical | C2×#41×C83 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C22×C4×C83 ◁ G | |
258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1826 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C83×#1882, C83×#1902, C83×#1905, … |
| maximal normal subgroup of | C83×#1882, C83×#1902, C83×#1905, … |
| maximal quotient of | C2×#419×C83, C2×#417×C83, C2×#428×C83, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 255);