Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149889 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | #3487 | C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,7,3,15,18)(4,9,20,28,6,16,26,23)(8,14,25,32,21,13,29,31)(10,27,19,24,17,22,12,30);(1,3)(2,7)(4,9)(5,23)(6,16)(8,13)(10,29)(11,30)(12,22)(14,21)(15,28)(17,25)(18,24)(19,27)(20,32)(26,31);(1,4,8,22)(2,9,14,12)(3,16,13,19)(5,20,25,30)(6,21,27,7)(10,11,28,32)(15,26,29,24)(17,18,23,31)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=ef,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57086);