Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C41 · Schur multiplier (H2) = C24 |
| direct product | C2×C41×#324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C41 | C22⋊C4 |
| commutator | C22 | C22×C8×C41 |
| Frattini, cosocle | C22×C4 | C23×C41 |
| prime radical | C24×C41 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C41 ◁ C23×C4×C41 ◁ G | |
328 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1804 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C41×#7189, C41×#7251, C4×C41×#324, … |
| maximal normal subgroup of | C41×#7189, C41×#7251, C4×C41×#324, … |
| maximal quotient of | C2×C41×#1271, C41×#7564, C41×#7573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 189);