Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | D4×#1889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×#120 |
| commutator | C24 | C24×C4 |
| Frattini, cosocle | C25 | C25 |
| prime radical | C2×D4×#120 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
29525 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D4×B(4,𝔽2), #72314030, #72314074, … |
| maximal normal subgroup of | C4×D4×B(4,𝔽2), #72314030, #72314074 |
| maximal quotient of | #72303174, #72312144, #72312659, … |
Past the SmallGroups library coverage — no SmallGroups id.