Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#147582 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #1781 |
| commutator | C23×C4 | C23×C4 |
| Frattini, cosocle | C22×#41 | C24 |
| prime radical | C2×#35527 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
16397 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72231756, #72280766, #72314507 |
| maximal normal subgroup of | C2×#72231756, #72280766, #72314507 |
| maximal quotient of | #72278567, #72281427, #72281596, … |
Past the SmallGroups library coverage — no SmallGroups id.