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 | |
4523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(26,31)(27,34)(33,38)(35,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,12)(10,16,17,18)(13,22,24,20)(14,21,19,23)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(2,6)(4,7)(10,17)(14,19)(16,18)(21,23)(29,37)(33,38)(35,40)(36,39);(2,6)(3,8)(9,12,15,11)(10,18,17,16)(13,20,24,22)(14,23,19,21)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,3)(2,7)(4,6)(5,8)(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,21)(18,23)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,4,5,7)(2,3,6,8)(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,22,18,20)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306164);