Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C2×#7622887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C26 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
35114 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
130 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72234840, C2×#72249654 |
| maximal normal subgroup of | C2×#72234840, C2×#72249654 |
| maximal quotient of | #72311217, #72311697, #72312163, … |
Past the SmallGroups library coverage — no SmallGroups id.