Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144538 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10351 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,4,9,6,8,13,11)(3,12,15,27,18,25,16,7)(5,20,19,28,23,29,22,10)(14,31,17,30,21,24,32,26);(1,3,13,15,6,18,4,16)(2,7,11,12,8,27,9,25)(5,21,22,32,23,14,19,17)(10,24,29,26,28,31,20,30);(2,8)(3,14)(4,13)(5,22)(7,24)(10,29)(12,30)(15,32)(16,17)(18,21)(19,23)(20,28)(25,26)(27,31)Polycyclic presentation:
2^9|a^2=d,b^2=dg,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51734);