Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13469 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2065 |
| 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 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12)(7,17,21,20)(8,10)(9,24)(13,31,33,34)(14,15)(16,28)(18,32,36,29)(19,35,22,26)(23,27);(5,6,8,12,11,16,10,28)(7,18,19,34,21,36,22,31)(9,25,23,14,27,30,24,15)(13,26,32,20,33,35,29,17);(2,4)(5,7)(6,13,16,33)(8,22)(9,26)(10,19)(11,21)(12,29,28,32)(14,31,15,34)(17,23)(18,30,36,25)(20,24)(27,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250147, #72250157, #72250188, … |
| maximal normal subgroup of | #72250147, #72250157, #72250188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56007);