Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,18,15,20,12,16)(11,21,26,36,22,37,24,40)(13,27,28,35,29,31,23,19)(17,32,33,38,34,39,30,25);(1,2,3,5,4,6,7,8)(10,16,20,18)(11,22)(17,33,34,30)(19,31,35,27)(21,36,37,40)(24,26)(25,38)(32,39);(2,6)(5,8)(9,11,23,39,12,24,29,38,15,22,28,32,14,26,13,25)(10,17,27,36,18,33,35,37,20,34,31,40,16,30,19,21)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48910);