Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2085 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,9,18,12,19,11,15)(7,20,22,13,25,42,38,43)(8,26,28,51,31,63,39,53)(10,33,32,17,37,49,36,16)(14,44,47,30,48,35,46,29)(21,55,56,65,58,68,57,45)(23,60,59,52,24,41,34,50)(27,54,61,40,64,66,62,67);(5,7,9,22,12,25,11,38)(6,13,15,20,19,43,18,42)(8,27,28,61,31,64,39,62)(10,34,32,23,37,59,36,24)(14,45,46,68,48,65,47,55)(16,50,49,41,17,52,33,60)(21,35,56,29,58,44,57,30)(26,40,53,54,63,67,51,66);(2,4)(5,8,9,28,12,31,11,39)(6,14,15,46,19,48,18,47)(7,21,22,56,25,58,38,57)(10,35,32,29,37,44,36,30)(13,40,20,54,43,67,42,66)(16,51,49,26,17,53,33,63)(23,61,59,64,24,62,34,27)(41,55,52,45,60,68,50,65)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55976);