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 | |
4333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,13)(14,15)(17,22)(18,21)(19,23)(20,24)(26,30)(27,33)(32,38)(35,36);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,29,30)(27,34,33,31)(28,32,37,38)(35,40,36,39);(2,3)(6,7)(10,15)(12,22)(16,18)(19,24)(25,26,29,30)(27,31,33,34)(28,32,37,38)(35,39,36,40);(4,8)(6,7)(9,11)(10,16)(12,19)(13,23)(14,21)(15,18)(17,20)(22,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,28,29,37)(26,32,30,38)(27,35,33,36)(31,39,34,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=ghi,f^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, 8521387);