Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#72210066 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #2029 |
| commutator | C23×Q8 | C23×C4 |
| Frattini, cosocle | C24×Q8 | C24 |
| prime radical | C26 | #120 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ C25×Q8 ◁ G | |
| LECS | G ▷ C24×Q8 ▷ C23 ▷ C2 ▷ C1 | |
27548 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
152 conjugacy classes — view the full character table.
| maximal subgroup of | #72266500, C2×#72210082, C2×#72210085, … |
| maximal normal subgroup of | #72266500, #72266626 |
Past the SmallGroups library coverage — no SmallGroups id.