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 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,38)(35,39)(36,40);(2,6)(3,7)(10,14)(15,17)(16,20)(23,24)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(1,3)(2,4)(5,7)(6,8)(11,18)(15,17)(19,21)(23,24)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(1,4)(2,3)(5,8)(6,7)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748547);