Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C218 |
| direct product | C2×#72235589 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C23×C4 | C26 |
| socle | C25 | C24×C4 |
| prime radical | C26×D4 | C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
852912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(7,8,9,10)(11,12)(13,14)(15,16,17,21,19,20,18,22);(8,10)(16,20)(21,22);(3,4)(5,6)(12,14);(4,6)(16,20)(21,22);(16,20)(21,22)464 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (7,8,9,10)(11,12)(13,14)(15,16,17,21,19,20,18,22), (8,10)(16,20)(21,22), (3,4)(5,6)(12,14), (4,6)(16,20)(21,22), (16,20)(21,22));