Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(7,17)(8,22)(9,24)(10,23)(12,28)(13,14)(15,26)(18,34)(19,35)(20,21)(25,33)(27,36)(29,32)(30,31);(5,6)(7,18,19,31)(8,14)(9,25)(10,26)(11,16)(12,20,29,36)(13,22)(15,23)(17,30,35,34)(21,28,27,32)(24,33);(5,7,10,27,11,19,23,21)(6,12,15,34,16,29,26,30)(8,20,22,35,24,36,9,17)(13,31,14,32,25,18,33,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240833, #72250027, #72250032, … |
| maximal normal subgroup of | #72240833, #72250027, #72250032, … |
| maximal quotient of | #72240837 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55983);