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 | |
4583 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)(5,6)(7,8)(25,26,30,31)(27,29,33,38)(28,35,37,32)(34,40,39,36);(2,6)(3,7)(9,10,14,13)(11,17,19,15)(12,16,21,20)(18,23,24,22)(25,26,30,31)(27,29,33,38)(28,32,37,35)(34,36,39,40);(1,3,5,7)(2,4,6,8)(10,13)(15,17)(16,20)(22,23)(26,31)(29,38)(32,35)(36,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,35)(33,39)(38,40);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=h,e^2=ghi,f^2=ghi,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, 8104500);