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)(5,6,9,17,12,18,33,15)(7,19,21,13,24,43,23,44)(8,25,29,51,30,47,27,64)(10,34,31,16,11,37,32,52)(14,39,46,65,48,36,49,28)(20,55,58,45,59,67,56,68)(22,41,38,50,61,42,35,54)(26,62,63,66,57,53,60,40);(5,7,9,21,12,24,33,23)(6,13,15,19,18,44,17,43)(8,26,27,60,30,57,29,63)(10,35,31,22,11,38,32,61)(14,45,46,67,48,68,49,55)(16,50,34,41,52,54,37,42)(20,39,56,28,59,36,58,65)(25,40,51,62,47,66,64,53);(2,4)(5,8,12,30)(6,14,18,48)(7,20)(9,29,33,27)(10,36,11,39)(13,40)(15,49,17,46)(16,51,52,64)(19,53)(21,58)(22,60)(23,56)(24,59)(25,37,47,34)(26,38)(28,32,65,31)(35,57)(41,55)(42,67)(43,62)(44,66)(45,50)(54,68)(61,63)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,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=eh,f^b=fhi,h^b=hi,d^c=di,e^c=eh,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, 55998);