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 | #28247 | 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 | |
365 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 76 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)(5,6)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20);(1,2)(3,4)(5,6)(7,13)(8,12)(9,14)(10,18)(11,15)(16,20)(17,19);(1,3,4,2)(6,12,15,20)(7,16)(8,17)(9,10)(13,18,19,14)Polycyclic presentation:
2^9|c^2=dgh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cf,f^b=fh,h^b=hi,e^c=egi,f^c=fhi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217768 |
| maximal normal subgroup of | #72217768 |
| maximal quotient of | #72215348, #72215350, #72215352 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52351);