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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(10,14)(15,22)(16,19)(23,24)(25,26,31,29)(27,32,34,28)(30,33,39,37)(35,38,40,36);(2,7)(3,4)(9,10,13,14)(11,15,18,22)(12,16,21,19)(17,23,20,24)(27,34)(28,32)(35,40)(36,38);(1,3,6,4)(2,8,7,5)(25,27,31,34)(26,32,29,28)(30,35,39,40)(33,38,37,36);(1,4,6,3)(2,5,7,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,28)(26,27)(29,34)(30,38)(31,32)(33,40)(35,37)(36,39);(1,5)(2,3)(4,7)(6,8)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,29,31,26)(27,28,34,32)(30,33,39,37)(35,38,40,36);(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,30,31,39)(26,33,29,37)(27,35,34,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=gi,f^2=gi,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, 8608915);