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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,3,6,5)(2,7,4,8)(10,15)(12,21)(16,19)(18,22)(27,36)(29,37)(32,35)(34,40);(9,11,14,20)(10,16,15,19)(12,18,21,22)(13,17,23,24)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(2,4)(7,8)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(1,4,6,2)(3,8,5,7)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,5,6,3)(2,7,4,8)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,30,31,26)(27,35,36,32)(28,38,39,33)(29,34,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123855);