Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1942 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
470 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)(5,6,9,18,12,19,35,15)(7,20,22,13,26,42,25,43)(8,27,29,62,33,52,32,46)(10,17,11,16,34,50,38,49)(14,31,47,37,48,64,45,30)(21,44,54,65,56,67,60,68)(23,40,24,59,36,51,58,41)(28,39,63,61,57,66,55,53);(5,7,9,22,12,26,35,25)(6,13,15,20,19,43,18,42)(8,28,29,63,33,57,32,55)(10,24,11,36,34,58,38,23)(14,44,45,68,48,67,47,65)(16,41,17,51,49,59,50,40)(21,37,54,64,56,30,60,31)(27,61,46,39,52,53,62,66);(2,4)(5,8,9,29,12,33,35,32)(6,14,15,45,19,48,18,47)(7,21,22,54,26,56,25,60)(10,31,11,37,34,64,38,30)(13,39,20,53,43,66,42,61)(16,46,17,52,49,62,50,27)(23,57,24,55,36,28,58,63)(40,67,41,65,51,44,59,68)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=h,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=fhi,h^b=hi,e^c=ehi,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, 55978);