Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C222 |
| direct product | C25×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | D4 |
| commutator | C4 | C26×C4 |
| Frattini, cosocle | C2×C4 | C27 |
| prime radical | C26×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ C26×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
505877 conjugacy classes of subgroups; only the 108373 normal subgroups are shown.
This group has 505877 conjugacy classes of subgroups — too many to draw. Only its 108373 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 — 7 generators on 22 points:
(1,2);(11,12,13,14)(16,19)(17,21)(20,22);(11,12)(13,14)(15,16)(17,22)(18,20)(19,21);(3,4);(5,6);(7,8);(9,10)448 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C25×#442, C2×#72237130, C2×#72237125, … |
| maximal normal subgroup of | C25×#442, C2×#72237130, C2×#72237125, … |
| maximal quotient of | C26×#102 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (11,12,13,14)(16,19)(17,21)(20,22), (11,12)(13,14)(15,16)(17,22)(18,20)(19,21), (3,4), (5,6), (7,8), (9,10));