Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,14,18,15,11,20,21)(12,19,22,24,16,17,13,23);(9,11,12,17,15,10,16,19)(13,21,20,24,22,18,14,23);(2,5)(6,8)(9,12,15,16)(10,17,11,19)(13,14,22,20)(18,23,21,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dh,e^b=efi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213264, #72213302, #72213311, … |
| maximal normal subgroup of | #72213264, #72213302, #72213311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57028);