Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C2×#511819 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C25 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
24665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,9)(10,13)(11,17)(16,18);(4,10)(5,7)(9,13)(11,17)(12,14)(16,18);(4,11,13,18)(5,7)(9,16,10,17)(12,14);(3,4,12,16,8,13,14,17)(5,9,15,18,7,10,6,11)106 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (4,9)(10,13)(11,17)(16,18), (4,10)(5,7)(9,13)(11,17)(12,14)(16,18), (4,11,13,18)(5,7)(9,16,10,17)(12,14), (3,4,12,16,8,13,14,17)(5,9,15,18,7,10,6,11));