Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | H(3,ℤ/4) |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C4×M32 | C22 |
| socle | C2 | #7146 |
| prime radical | C23 | #364 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,6,14,27,40,7,12,28,41,25,43,26,55,8,13,29,42,30,10,35,37,23,38,54,57,4,21,24,53)(3,15,17,44,18,33,49,59,19,45,50,63,16,34,47,39,20,36,51,58,48,60,61,62,46,52,56,64,22,9,31,32);(1,3)(2,9,12,33,13,34,38,60)(4,22)(5,17)(6,18)(7,19)(8,20)(10,36,21,52,14,15,43,45)(11,39,41,62,42,32,57,59)(16,25)(23,46)(24,31)(26,47)(27,49)(28,50)(29,51)(30,48)(35,61)(37,44,53,63,40,58,55,64)(54,56)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=ch,e^a=efh,f^a=fi,c^b=ci,d^b=df,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1205);