Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1925 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
456 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)(6,11,14,20)(12,26,27,28)(13,23,30,35)(15,29,31,32);(1,2,4,3)(5,6)(7,15)(8,20)(9,23)(10,14)(11,22)(12,17)(13,21)(16,32)(18,29)(19,31)(24,28)(25,35)(26,33)(27,34)(30,36);(1,3,4,2)(5,7,8,16,10,19,22,18)(6,12,11,26,14,27,20,28)(9,24,21,34,25,33,36,17)(13,29,23,31,30,32,35,15)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fh,g^b=gi,h^b=hi,f^c=fhi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56144);