Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×#493522 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×#102 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| socle | C23 | C22×#102 |
| prime radical | C24×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ C25×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
32269 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
184 conjugacy classes — view the full character table.
| maximal subgroup of | #72320354, #72320768, #72320892 |
| maximal normal subgroup of | #72320354, #72320768, #72320892 |
Past the SmallGroups library coverage — no SmallGroups id.