Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | C23 | C8×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,15,21,13,20,29,38,16,22,32,39,14,19,31,37)(11,23,25,27,18,34,35,28,26,36,40,30,24,17,33,12);(1,2)(3,4)(5,7)(6,8)(9,11,15,25,13,18,29,35,16,26,32,40,14,24,31,33)(10,17,21,12,20,23,38,27,22,34,39,28,19,36,37,30);(1,3,6,7)(2,4,8,5)(9,12,16,28)(10,18,22,24)(11,20,26,19)(13,27,14,30)(15,23,32,36)(17,29,34,31)(21,35,39,33)(25,38,40,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=egi,d^c=dgi,e^c=eg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43093);