Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C219 |
| direct product | C2×#72251006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 12222 conjugacy classes of subgroups; only the 23304 normal subgroups are shown.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 22 points:
(7,8)(9,10)(11,12)(13,14)(15,16,18,17)(20,22);(4,6)(8,10)(12,14)(16,17)(19,20)(21,22);(12,14)(20,22);(19,20)(21,22);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,16)392 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((7,8)(9,10)(11,12)(13,14)(15,16,18,17)(20,22), (4,6)(8,10)(12,14)(16,17)(19,20)(21,22), (12,14)(20,22), (19,20)(21,22), (1,2), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,16));