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 | |
4411 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)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(25,26)(27,34)(28,36)(29,32)(30,33)(31,37)(35,39)(38,40);(2,6)(3,7)(10,15)(11,20)(17,22)(18,21)(25,27,30,32)(26,29,33,34)(28,35,37,40)(31,39,36,38);(1,3,5,7)(2,4,6,8)(10,15)(11,20)(17,22)(18,21)(28,37)(31,36)(35,40)(38,39);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,28,30,37)(26,31,33,36)(27,35,32,40)(29,39,34,38);(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,26)(27,29)(28,31)(30,33)(32,34)(35,39)(36,37)(38,40);(9,13,14,24)(10,18,15,21)(11,17,20,22)(12,19,23,16)(25,29)(26,32)(27,33)(28,38)(30,34)(31,35)(36,40)(37,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627733);