Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#72216427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #97550 |
| commutator | C26 | C23×C4 |
| Frattini, cosocle | C27 | C24 |
| prime radical | C2×#6499867 | C2 |
| LCS | G ▷ C26 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C28 ◁ G | |
| LECS | G ▷ C27 ▷ C24 ▷ C2 ▷ C1 | |
79528 conjugacy classes of subgroups; only the 472 normal subgroups are shown.
This group has 79528 conjugacy classes of subgroups — too many to draw. Only its 472 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
128 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#72216927 |
| maximal subgroup of | C2×#72216927, C2×#72216433, C2×#72216490, … |
| maximal normal subgroup of | C2×#72216927, C2×#72216433, C2×#72216490, … |
Past the SmallGroups library coverage — no SmallGroups id.