Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C25×#123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C23 |
| commutator, Frattini, cosocle | C22 | C28 |
| prime radical | C26×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
1138677 conjugacy classes of subgroups; only the 469973 normal subgroups are shown.
This group has 1138677 conjugacy classes of subgroups — too many to draw. Only its 469973 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 22 points:
(16,19)(20,22);(12,14)(15,16,18,19)(17,20,21,22);(3,4);(11,12)(13,14)(15,17,18,21)(16,20,19,22);(5,6);(7,8);(9,10);(1,2)(3,4)(5,6)(7,8)(9,10)448 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C24×#2430, C25×#546, C24×#2613, … |
| maximal normal subgroup of | C24×#2430, C25×#546, C24×#2613, … |
| maximal quotient of | C24×#2405, C24×#2849, C22×#7630942, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((16,19)(20,22), (12,14)(15,16,18,19)(17,20,21,22), (3,4), (11,12)(13,14)(15,17,18,21)(16,20,19,22), (5,6), (7,8), (9,10), (1,2)(3,4)(5,6)(7,8)(9,10));