Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C222 |
| direct product | C24×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | #120 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
433653 conjugacy classes of subgroups; only the 38709 normal subgroups are shown.
This group has 433653 conjugacy classes of subgroups — too many to draw. Only its 38709 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
256 conjugacy classes — view the full character table.
| maximal subgroup of | #72240591, C22×#394066, C22×#498927, … |
| maximal normal subgroup of | #72240591, C22×#394066, C22×#498927, … |
| maximal quotient of | C2×#72216616, C23×#33551, C22×#7622880, … |
Past the SmallGroups library coverage — no SmallGroups id.