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 | C2×C4 | #476 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1987 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,28,34,26,14,11,16,25,36,10,17,8,24)(7,18,21,12,29,32,19,13,23,35,20,31,22,30,27,33);(1,2)(3,4)(5,7)(6,12,16,31)(8,21)(9,27)(10,29)(11,23)(13,34,33,17)(14,30,24,32)(15,18,36,35)(19,25)(20,26)(22,28);(2,4)(5,7)(6,13,16,33)(8,21)(9,27)(10,29)(11,23)(12,17,31,34)(14,18,24,35)(15,32,36,30)(19,25)(20,26)(22,28)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56547);