Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#72235853 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #8545 |
| commutator | C22×C42 | C23×C4 |
| Frattini, cosocle | C22×#126 | C24 |
| prime radical | C26 | #120 |
| LCS | G ▷ C22×C42 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×#126 ◁ G | |
| LECS | G ▷ C22×#126 ▷ C24 ▷ C22 ▷ C1 | |
9434 conjugacy classes of subgroups; only the 376 normal subgroups are shown.
This group has 9434 conjugacy classes of subgroups — too many to draw. Only its 376 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
176 conjugacy classes — view the full character table.
| maximal subgroup of | #72235860, C2×#72235878, C2×#72235894 |
| maximal normal subgroup of | #72235860 |
Past the SmallGroups library coverage — no SmallGroups id.