Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#512900 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2196 |
| commutator, Frattini, cosocle | C4⋊Q8 | C25 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
7822 conjugacy classes of subgroups.
| transitive degree | μt(G) = 32 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72231248, C2×#72231245, C2×#72231252, … |
| maximal normal subgroup of | C2×#72231248, C2×#72231245, C2×#72231252, … |
Past the SmallGroups library coverage — no SmallGroups id.