Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1634 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10056 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,12,16,30)(4,18)(5,10)(6,23)(7,11)(8,15,25,31)(9,21)(13,19,24,32)(14,29,17,26)(20,27)(22,28);(1,2)(3,13,16,24)(4,18)(5,10)(6,23)(7,11)(8,14,25,17)(9,21)(12,15,30,31)(19,26,32,29)(20,27)(22,28);(1,3)(2,8)(4,19)(5,17)(6,14)(7,16)(9,26)(10,24)(11,25)(12,28)(13,23)(15,20)(18,30)(21,31)(22,32)(27,29)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50522);