Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,10,12,20)(7,14)(8,17)(11,19,13,15)(16,18);(1,3,4,2)(5,6)(7,15)(8,11)(9,12)(10,18)(13,17)(14,20)(16,19);(7,14)(8,17)(10,19)(11,13)(15,20)(16,18)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217755, #72221570, #72226784, … |
| maximal normal subgroup of | #72217755, #72226784, #72226787, … |
| maximal quotient of | #72217001, #72217006, #72217009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43731);