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 | |
4582 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)(3,4)(5,8)(6,7)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,32,35)(27,36,38,33)(28,31,29,39)(30,34,40,37);(1,3)(2,4)(5,6)(7,8)(9,11,14,18)(10,15,17,22)(12,19,16,23)(13,20,21,24)(25,27,32,38)(26,33,35,36)(28,34,29,37)(30,39,40,31);(1,4)(2,5)(3,7)(6,8)(11,18)(15,22)(19,23)(20,24)(25,28,32,29)(26,31,35,39)(27,34,38,37)(30,36,40,33);(1,3,6,5)(2,4,7,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,29,32,28)(26,31,35,39)(27,37,38,34)(30,36,40,33);(1,5,6,3)(2,4,7,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30)(26,34)(27,31)(28,33)(29,36)(32,40)(35,37)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,31)(26,29)(27,30)(28,35)(32,39)(33,37)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8503625);