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 | |
4486 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,22)(23,24)(25,26,31,28)(27,32,35,36)(29,33,39,38)(30,37,40,34);(2,6)(3,7)(9,11,14,16)(10,13,17,18)(12,19,21,23)(15,22,20,24)(25,27)(26,32)(28,36)(29,34)(30,33)(31,35)(37,39)(38,40);(1,3,5,7)(2,4,6,8)(12,21)(15,20)(19,23)(22,24)(26,28)(27,35)(33,38)(34,37);(1,4)(2,7)(3,6)(5,8)(9,12)(10,15)(11,19)(13,22)(14,21)(16,23)(17,20)(18,24)(25,28,31,26)(27,36,35,32)(29,33,39,38)(30,37,40,34);(9,10)(11,13)(12,20)(14,17)(15,21)(16,18)(19,24)(22,23)(25,29)(26,33)(27,37)(28,38)(30,36)(31,39)(32,40)(34,35);(9,13)(10,16)(11,17)(12,22)(14,18)(15,23)(19,20)(21,24)(25,30)(26,34)(27,33)(28,37)(29,32)(31,40)(35,38)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050543);