Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#153461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13561 |
| commutator | #127 | C23×C4 |
| Frattini, cosocle | #493 | C24 |
| prime radical | C2×#18524 | C2 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C22×C42 ◁ C2×#3151 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3476 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 3476 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(3,4)(5,13,15,14,16,18,17,10)(6,9)(7,12)(8,11);(3,5,6,14,8,17,11,13,9,16,4,10,7,15,12,18);(1,2);(4,6)(10,14)(11,12)(13,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #72279242, C2×#72231241, C2×#72231263, … |
| maximal normal subgroup of | #72279242, C2×#72231241, C2×#72231263, … |
| maximal quotient of | #72280016 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,13,15,14,16,18,17,10)(6,9)(7,12)(8,11), (3,5,6,14,8,17,11,13,9,16,4,10,7,15,12,18), (1,2), (4,6)(10,14)(11,12)(13,18));