Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216 |
| direct product | C22×#501097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C26 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ C25×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
107832 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
188 conjugacy classes — view the full character table.
| maximal subgroup of | #72285761, #72318574 |
| maximal normal subgroup of | #72285761, #72318574 |
| maximal quotient of | #72319331 |
| automorphism group of | #7372, #7402, #7403, … |
Past the SmallGroups library coverage — no SmallGroups id.