Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×#493635 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #20609 |
| commutator | C2×D4 | C24×C4 |
| Frattini, cosocle | C22×D4 | C25 |
| prime radical | C2×#63011 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
23824 conjugacy classes of subgroups.
| transitive degree | μt(G) = 32 |
124 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72216366, #72249557, #72249579, … |
| maximal normal subgroup of | C2×#72216366, #72249557, #72249579, … |
| maximal quotient of | #72249557, #72255810, #72255994, … |
Past the SmallGroups library coverage — no SmallGroups id.