Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C224 |
| direct product | C24×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C23 |
| commutator | C22 | C26×C4 |
| Frattini, cosocle | C23 | C27 |
| prime radical | C29 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324501 conjugacy classes of subgroups; only the 156245 normal subgroups are shown.
This group has 4324501 conjugacy classes of subgroups — too many to draw. Only its 156245 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 20 points:
(3,4);(5,6);(7,8);(10,12)(13,14,15,16)(18,20);(1,2)(3,4)(5,6)(7,8);(17,18)(19,20);(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)448 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C24×#2846, C2×#72236365, C23×#14679, … |
| maximal normal subgroup of | C24×#2846, C2×#72236365, C23×#14679, … |
| maximal quotient of | C23×#14679, C22×D4×#379, C23×#13867, … |
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), (10,12)(13,14,15,16)(18,20), (1,2)(3,4)(5,6)(7,8), (17,18)(19,20), (9,10)(11,12)(13,14)(15,16)(17,18)(19,20));