Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212×C4 |
| direct product | C22×#8546 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #379 |
| commutator | C24 | C24×C4 |
| Frattini, cosocle | C25 | C25 |
| prime radical | C24×#127 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C27 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
70501 conjugacy classes of subgroups; only the 2229 normal subgroups are shown.
This group has 70501 conjugacy classes of subgroups — too many to draw. Only its 2229 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,9,10,12)(7,11)(14,17,18,20)(15,19);(13,14,16,18)(15,17,19,20);(1,2)(6,10)(9,12)(14,18)(17,20);(3,4);(5,6,8,10)(7,9,11,12)160 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#245233 |
| maximal normal subgroup of | C22×#245233 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,9,10,12)(7,11)(14,17,18,20)(15,19), (13,14,16,18)(15,17,19,20), (1,2)(6,10)(9,12)(14,18)(17,20), (3,4), (5,6,8,10)(7,9,11,12));