Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143324 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28504 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,3)(7,15)(8,17)(10,12)(11,14)(16,18)(19,20);(1,2)(3,4)(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,20)(18,19);(1,3,4,2)(6,10,13,19)(7,16)(8,17)(11,20,14,12)(15,18)Polycyclic presentation:
2^9|c^2=dg,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eg,f^c=fhi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217764, #72226863, #72226869, … |
| maximal normal subgroup of | #72217764, #72226863, #72226869, … |
| maximal quotient of | #72216332, #72216351, #72216354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52213);