Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C217 |
| direct product | C2×#6628897 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C26 |
| prime radical | C22×H(7,𝔽2) | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
229329 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
184 conjugacy classes — view the full character table.
| maximal subgroup of | #72252225, #72252230, #72252231, … |
| maximal normal subgroup of | #72252225, #72252230, #72252231, … |
Past the SmallGroups library coverage — no SmallGroups id.