Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#107 |
| prime radical | C23 | C4×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2,5,7,4,6,8,3)(9,10,13,19,15,21,31,38,16,22,30,37,14,20,12,18)(11,23,26,35,27,36,25,32,28,39,29,17,24,33,40,34);(2,6)(3,7)(9,11,14,27,16,28,15,24)(10,17,20,34,22,35,21,32)(12,29,30,40,31,26,13,25)(18,36,37,39,38,33,19,23);(1,3,8,6,4,7,5,2)(11,24,28,27)(17,32,35,34)(23,33,39,36)(25,26,40,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^b=ch,d^b=dg,e^b=eg,g^b=gi,h^b=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36352);