Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1624 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10413 |
| 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 | |
281 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,7,11)(3,12)(4,5,9,17)(6,18,22,26)(8,19)(10,27,29,20)(13,30)(14,23)(15,25)(16,24)(21,31)(28,32);(1,3)(2,8)(4,12)(5,19)(6,21)(7,15)(9,25)(10,28)(11,23)(13,18)(14,17)(16,27)(20,30)(22,32)(24,26)(29,31);(1,4,17,11,7,9,5,2)(3,13,14,31,15,24,19,28)(6,10,27,18,22,29,20,26)(8,21,12,16,23,32,25,30)Polycyclic presentation:
2^9|a^2=g,c^2=dg,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=eg,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, 52257);