Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C4×#33561 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C24×C4 |
| Frattini, cosocle | C22×D4 | C25 |
| socle | C22 | C22×B(4,𝔽2) |
| prime radical | C2×#3018 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
7250 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#7622874, #72315023, #72315043, … |
| maximal normal subgroup of | C4×#7622874, #72315023, #72315043, … |
Past the SmallGroups library coverage — no SmallGroups id.