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 | |
441 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 441 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20,22,15)(12,16)(13,18,26,17)(14,19)(21,33,34,27)(23,32,36,28)(24,30,37,29)(25,31)(35,39,40,38);(9,11,13,22,14,23,26,36)(10,15,18,20,19,28,17,32)(12,21,24,34,25,35,37,40)(16,27,30,33,31,38,29,39);(2,5)(6,8)(9,12,14,25)(10,16,19,31)(11,21,23,35)(13,24,26,37)(15,27,28,38)(17,29,18,30)(20,33,32,39)(22,34,36,40)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,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57026);