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 | |
4589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,20)(16,21)(22,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(26,27)(31,33)(32,34)(38,39);(2,6)(4,7)(12,19)(16,21)(17,23)(22,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090126);