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 | |
4490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,38)(39,40);(1,3,6,5)(2,7,8,4)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,38)(36,37)(39,40);(1,3,6,5)(2,7,8,4)(12,21)(15,20)(18,24)(22,23)(26,31)(27,32)(33,40)(35,37);(2,8)(3,5)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24)(25,27,30,32)(26,28,31,34)(29,37,39,35)(33,38,40,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,5,6,3)(2,7,8,4)(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,29,30,39)(26,33,31,40)(27,35,32,37)(28,36,34,38)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,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, 8627997);