Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C218×C4 |
| direct product | C2×#72298516 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C25×#127 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
1069128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 22 points:
(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(16,18)(20,22);(15,16)(17,18)(19,20)(21,22);(4,6)(7,8,9,10)(12,14)(16,18);(4,6)(8,10)(11,12,13,14)(15,16)(17,18);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14);(1,2)368 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((3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(16,18)(20,22), (15,16)(17,18)(19,20)(21,22), (4,6)(7,8,9,10)(12,14)(16,18), (4,6)(8,10)(11,12,13,14)(15,16)(17,18), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14), (1,2));