Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#147496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C42 |
| Frattini, cosocle | C23×Q8 | C24 |
| prime radical | C23×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
12173 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
160 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#72210824 |
| maximal subgroup of | #72267301, C2×#72210824, #72267420, … |
| maximal normal subgroup of | #72267301, C2×#72210824, #72267420, … |
Past the SmallGroups library coverage — no SmallGroups id.