Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| 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 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,7)(10,15,17,23)(11,12)(13,20)(16,18,24,22)(19,21);(5,6,8,7)(10,16,17,24)(11,12)(13,20)(15,22,23,18)(19,21);(2,4)(5,7,8,6)(9,10,13,22,14,17,20,18)(11,16,12,15,19,24,21,23)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=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250461, #72250666, #72250668 |
| maximal normal subgroup of | #72250461, #72250666, #72250668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55981);