Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#147567 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #1781 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | C2×#34980 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C25×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
21871 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213329, C22×#147567, D3×#147567, … |
| maximal normal subgroup of | C2×#72213329, C22×#147567, #72250046, … |
| maximal quotient of | #72236641, C2×#72213256, #72236699, … |
Past the SmallGroups library coverage — no SmallGroups id.