Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149895 |
| 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, prime radical | C22×D4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(1,2,5,9,6,10,4,8)(3,7,11,14,12,15,13,16)(18,22,23,31,24,26,28,32)(19,25,27,29);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(17,18,20,28,21,24,30,23)(19,26,25,32,27,22,29,31);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=gh,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cgh,d^b=dh,e^b=efi,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, 57230);