Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C41 · Schur multiplier (H2) = C27 |
| direct product | C22×#102×C41 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C41 | D4 |
| commutator | C4 | C23×C4×C41 |
| Frattini, cosocle | C2×C4 | C24×C41 |
| prime radical | C23×D4×C41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C41 ◁ C23×C4×C41 ◁ G | |
1072 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2296 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C41×#10788, C41×#10925, C41×#10927, … |
| maximal normal subgroup of | C41×#10788, C41×#10925, C41×#10927, … |
| maximal quotient of | C22×C41×D4⋊C8, C2×C41×#1486, C2×C41×#1480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 1622);