Finite Group DatabasebetaFinite Group Databasebeta| order | 5248 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C41 · Schur multiplier (H2) = C22×C4 |
| direct product | C41×#2199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C41 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C41 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C41 ◁ C22×C41 ◁ C2×D4×C41 ◁ G | |
270 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
697 conjugacy classes — view the full character table.
| maximal subgroup of | C41×#13677, C41×#13678, C2×C41×#2199, … |
| maximal normal subgroup of | C41×#13677, C41×#13678, C2×C41×#2199, … |
| maximal quotient of | C41×#12093, C41×#12100, C41×#12123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5248, 931);