Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141376 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9764 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #405 | C23 |
| prime radical | #3176 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,13,35,11,8,15,36,46,37,14,47,29)(3,5,18,16,19,32,34,51,21,33,45,31,20,50,10,43)(4,22,26,9,27,39,58,40,28,59,25,42,62,48,63,41)(17,52,54,64,23,60,44,24,56,49,57,61,55,53,38,30);(1,3,16,13,37,19,43,15,8,21,31,14,7,20,51,2)(4,23,49,42,62,56,60,41,28,55,52,9,27,17,53,40)(5,29,6,34,50,12,47,45,33,11,36,10,32,46,35,18)(22,58,44,24,39,26,57,64,59,63,38,30,48,25,54,61);(1,4,6,25,7,27,35,63,8,28,36,26,37,62,47,58)(2,9,11,22,14,41,12,48,15,42,29,59,13,40,46,39)(3,17,10,44,20,55,45,54,21,56,34,38,19,23,18,57)(5,30,31,52,32,64,43,60,33,24,16,49,50,61,51,53)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=ghi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48572);