Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,13)(14,16)(15,21)(18,23)(19,22)(20,24)(26,29)(31,33)(32,37)(38,39);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,16,17)(12,19,18,24)(13,20,23,22)(28,36)(32,37)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,16)(12,18)(15,17)(19,24)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,12,14,18)(10,13,16,23)(11,19,21,24)(15,20,17,22)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(9,11)(10,17)(12,19)(13,22)(14,21)(15,16)(18,24)(20,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,13)(10,18)(11,20)(12,16)(14,23)(15,24)(17,19)(21,22)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620936);