Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C41 · Schur multiplier (H2) = C24 |
| direct product | Q8×Q16×C41 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C41 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C4×Q8×C41 ◁ G | |
336 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1435 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C41×#12354, C41×#12358, C41×#12406, … |
| maximal normal subgroup of | C41×#12354, C41×#12358, C41×#12406, … |
| maximal quotient of | Q8×#103×C41, C4⋊C4×Q16×C41, Q8×#107×C41, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 2114);