Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6,7,8)(11,18)(14,19)(16,24)(20,21);(1,2,4,3)(9,10,12,22,13,15,23,17)(11,19,21,16,18,14,20,24);(1,3,4,2)(10,14)(11,20)(12,23)(15,19)(16,22)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56154);