Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C2×#7622880 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C26 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
38452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(8,13)(11,14);(4,8,10,13)(5,11,12,14);(3,4)(5,9)(6,13)(7,10)(8,15)(11,16)(12,17)(14,18);(3,5,7,12)(4,9,10,17)(6,14,15,11)(8,16,13,18);(4,10)(8,13)(9,17)(16,18)136 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), (8,13)(11,14), (4,8,10,13)(5,11,12,14), (3,4)(5,9)(6,13)(7,10)(8,15)(11,16)(12,17)(14,18), (3,5,7,12)(4,9,10,17)(6,14,15,11)(8,16,13,18), (4,10)(8,13)(9,17)(16,18));