Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136885 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #448 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,16)(4,6)(5,8)(7,10,14,13)(12,15)(17,18,21,25,31,29,20,30,23,26,32,28,22,19,27,24);(2,6)(5,12)(7,14)(10,13)(17,19,22,18,23,29,31,26)(20,30,21,28,27,24,32,25);(1,3,4,11)(2,7,6,14)(5,13,12,10)(8,16,15,9)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,c^2=h,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44201);