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 | |
4281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26,31,33)(27,36,38,32)(28,34,39,29)(30,37,40,35);(1,3)(2,6)(4,8)(5,7)(9,11,15,20)(10,16,17,24)(12,21,23,14)(13,19,18,22)(25,27,31,38)(26,32,33,36)(28,37,39,35)(29,40,34,30);(10,17)(11,20)(12,23)(19,22)(26,33)(28,39)(32,36)(35,37);(9,12)(10,18)(11,21)(13,17)(14,20)(15,23)(16,22)(19,24)(25,28)(26,34)(27,37)(29,33)(30,32)(31,39)(35,38)(36,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,18)(10,12,17,23)(11,22,20,19)(14,24,21,16)(25,29,31,34)(26,28,33,39)(27,30,38,40)(32,35,36,37);(1,4)(2,7)(3,8)(5,6)(9,14,15,21)(10,19,17,22)(11,12,20,23)(13,16,18,24)(25,30)(26,35)(27,29)(28,32)(31,40)(33,37)(34,38)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=h,f^2=i,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, 9623529);