Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | #102×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24×D4 |
| commutator | C2×C4 | C25×C4 |
| Frattini, cosocle | C22×C4 | C26 |
| prime radical | C2×D4×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
43019 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
238 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×#442, #72313280, #102×B(4,𝔽2) |
| maximal normal subgroup of | H(5,𝔽2)×#442, #72313280, #102×B(4,𝔽2) |
| maximal quotient of | #72315551, #72315974 |
Past the SmallGroups library coverage — no SmallGroups id.