Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#147787 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #1790 |
| commutator | C22×Q8 | C23×C4 |
| Frattini, cosocle | C22×#41 | C24 |
| prime radical | C2×#37297 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
8925 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72214565, D3×#147787, #72261920, … |
| maximal normal subgroup of | C2×#72214565, #72261920, C2×#72231756 |
| maximal quotient of | C2×#72214258, D3×#147787, #72278597 |
Past the SmallGroups library coverage — no SmallGroups id.