Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10481 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| 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 | |
258 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,8,12)(3,15)(4,19)(5,13,20,28)(6,25,24,11)(7,27,26,14)(9,22)(10,17)(16,31)(18,32)(21,30)(23,29);(1,3)(2,9)(4,6)(5,17)(7,23)(8,18)(10,14)(11,31)(12,19)(13,32)(15,28)(16,26)(20,21)(22,24)(25,29)(27,30);(1,4,20,16,7,21,6,18,8,22,5,23,26,17,24,3)(2,10,25,31,14,9,28,15,12,30,11,29,27,19,13,32)Polycyclic presentation:
2^9|a^2=h,c^2=dh,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dgi,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=eghi,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52642);