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 | #2101 |
| 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 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,11,18,13,19,12,39)(7,20,25,42,26,51,24,53)(8,27,31,48,33,58,32,63)(9,34,10,38,37,17,36,16)(14,40,44,35,45,23,43,22)(15,46,49,62,50,29,60,30)(21,41,56,65,57,66,28,47)(52,54,64,61,68,59,67,55);(1,2)(3,4)(5,7,9,35,11,25,10,23,13,26,37,22,12,24,36,40)(6,14,16,51,39,43,17,42,19,45,38,20,18,44,34,53)(8,28,29,61,31,21,30,59,33,56,46,55,32,57,62,54)(15,47,48,64,60,66,27,52,50,65,63,67,49,41,58,68);(2,4)(5,8,9,29,11,31,10,30,13,33,37,46,12,32,36,62)(6,15,16,48,39,60,17,27,19,50,38,63,18,49,34,58)(7,21,22,54,25,56,40,61,26,57,35,59,24,28,23,55)(14,41,20,52,43,47,53,67,45,66,51,68,44,65,42,64)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,f^b=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56209);