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 | |
4626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,17)(13,20)(14,15)(18,23)(19,22)(21,24)(25,26)(27,28)(29,37)(30,32)(31,34)(33,38)(35,39)(36,40);(1,3,5,8)(2,6,4,7)(10,15)(12,21)(16,22)(18,20)(25,27,30,34)(26,31,32,28)(29,35,38,40)(33,39,37,36);(1,2,5,4)(3,6,8,7)(10,15)(11,19)(17,24)(18,20)(26,32)(27,34)(29,38)(36,39);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,20)(13,17)(14,19)(15,22)(18,21)(23,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,39)(36,40)(37,38);(9,12,14,21)(10,17,15,24)(11,20,19,18)(13,22,23,16)(25,28,30,31)(26,27,32,34)(29,36,38,39)(33,35,37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7908208);