Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#128062 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8101 |
| commutator | C24 | C22×C42 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | C23×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
12009 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
106 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72238010, C2×#72238013, C2×#72213523, … |
| maximal normal subgroup of | C2×#72238010, C2×#72238013, C2×#72213523, … |
Past the SmallGroups library coverage — no SmallGroups id.