Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×#493391 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1781 |
| commutator | C24 | C24×C4 |
| Frattini, cosocle | C22×D4 | C25 |
| prime radical | C2×#62842 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
33445 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
130 conjugacy classes — view the full character table.
| maximal subgroup of | #72249360, #72251657, #72251660, … |
| maximal normal subgroup of | #72249360, #72251657, #72251660, … |
| maximal quotient of | #72249574, #72249577 |
Past the SmallGroups library coverage — no SmallGroups id.