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 |
| prime radical | #44935 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4453 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)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,34)(28,36)(29,31)(30,32)(33,38)(35,39)(37,40);(2,5)(6,8)(9,11,14,18)(10,13,15,17)(12,20,22,19)(16,21,24,23)(25,27,31,30)(26,32,29,34)(28,37,38,35)(33,40,36,39);(1,2,4,5)(3,6,7,8)(10,15)(11,18)(16,24)(19,20)(26,29)(32,34)(33,36)(39,40);(3,7)(6,8)(9,12,14,22)(10,16,15,24)(11,19,18,20)(13,23,17,21)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,19)(13,15)(14,18)(16,21)(20,22)(23,24)(25,29,31,26)(27,32,30,34)(28,36,38,33)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,17)(10,11,15,18)(12,21,22,23)(16,20,24,19)(25,30,31,27)(26,32,29,34)(28,35,38,37)(33,40,36,39)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gh,e^2=g,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, 8508305);