Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×C4×#1790 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | #120 |
| commutator | C23 | C23×C42 |
| Frattini, cosocle | C25 | C25 |
| socle | C24 | C2×#120 |
| prime radical | C23×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
40489 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
256 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#32898 |
| maximal normal subgroup of | C2×C4×#32898 |
| maximal quotient of | C2×#72317371 |
Past the SmallGroups library coverage — no SmallGroups id.