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 | |
4627 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,21)(13,23)(14,16)(15,17)(19,24)(20,22)(25,26)(27,33)(28,31)(29,34)(30,36)(32,37)(35,39)(38,40);(2,5)(6,7)(9,11,15,14)(10,16,17,18)(12,22,23,19)(13,20,21,24)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40);(1,2)(3,6)(4,5)(7,8)(10,17)(12,23)(16,18)(19,22)(25,28,29,37)(26,31,34,32)(27,35,36,40)(30,38,33,39);(1,3,4,8)(2,6,5,7)(9,12)(10,13)(11,19)(14,22)(15,23)(16,24)(17,21)(18,20)(25,28)(26,32)(27,35)(29,37)(30,39)(31,34)(33,38)(36,40);(9,13)(10,12)(11,20)(14,24)(15,21)(16,22)(17,23)(18,19)(25,27)(26,33)(28,35)(29,36)(30,34)(31,39)(32,38)(37,40);(9,14,15,11)(10,16,17,18)(12,19,23,22)(13,20,21,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,38)(39,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=i,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, 8626152);