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 | #41961 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,13)(11,12)(18,21)(19,20);(2,6)(3,7)(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(26,31)(27,33)(32,39)(34,40);(1,3)(2,4)(5,7)(6,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,26,30,31)(27,29,33,38)(28,32,36,39)(34,37,40,35);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,39)(36,40);(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,29)(26,33)(27,31)(28,35)(30,38)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8602497);