Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C216 |
| direct product | C2×#392125 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#120 |
| commutator | C24 | C24×C4 |
| Frattini, cosocle | C25 | C25 |
| prime radical | C24×H(5,𝔽2) | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C27 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
237701 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
196 conjugacy classes — view the full character table.
| maximal subgroup of | #72240147, C2×#72237497, C2×#72219111, … |
| maximal normal subgroup of | #72240147, C2×#72237497, C2×#72225609, … |
| maximal quotient of | #72252098, C2×#72233438 |
Past the SmallGroups library coverage — no SmallGroups id.