Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2058 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,17,19,35)(8,14)(9,27)(10,30)(11,16)(12,20,32,28)(13,25)(15,29)(18,22,36,31)(21,34,24,33)(23,26);(1,2,4,3)(5,6)(7,18,19,36)(8,23)(9,13)(10,30)(11,16)(12,21,32,24)(14,26)(15,29)(17,31,35,22)(20,33,28,34)(25,27);(1,3,4,2)(5,7,10,31,11,19,29,22)(6,12,15,34,16,32,30,33)(8,24,25,20,26,21,9,28)(13,18,14,17,27,36,23,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240833, #72249926, #72249941, … |
| maximal normal subgroup of | #72240833, #72249926, #72249941, … |
| maximal quotient of | #72240836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55989);