Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149899 |
| 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 | |
426 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 426 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,23,25,32,26,28,24,31)(20,27,29,30);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,24)(20,28)(21,25)(22,26)(23,29)(27,32)(30,31);(3,9)(6,12)(10,15)(13,16)(17,19,22,21)(18,24,26,25)(20,27,29,30)(23,31,28,32)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,c^b=cgh,d^b=di,e^b=ef,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57096);