Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151105 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
| affine general linear | AGL(1,ℤ/32) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12067 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| prime radical | #2263 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1313 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 138 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:
(2,7,9,23,11,25,21,26)(3,12,14,18)(4,16)(5,20)(8,17,27,10,28,29,24,30)(13,19)(15,22,32,31);(1,2)(3,8)(4,17)(5,21)(6,11)(7,22)(9,20)(10,13)(12,23)(14,28)(15,27)(16,30)(18,26)(19,29)(24,32)(25,31);(1,3,13,31,20,32,19,12,6,14,16,22,5,15,4,18)(2,8,10,25,9,24,29,23,11,28,30,7,21,27,17,26)Polycyclic presentation:
2^9|a^2=e,c^2=efhi,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fh,h^a=hi,d^b=dfhi,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58302);