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 | |
4359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(27,33)(30,37)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,19)(13,22)(18,20)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,4)(2,3)(5,6)(7,8)(9,11,14,17)(10,13,15,22)(12,20,19,18)(16,24,23,21)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9416159);