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 | #43022 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26)(27,30)(28,29)(31,34)(32,39)(33,40)(35,36)(37,38);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,27,31,33)(26,30,34,40)(28,37,39,35)(29,38,32,36);(2,6)(3,7)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,28,31,39)(26,32,34,29)(27,35,33,37)(30,38,40,36);(3,7)(4,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23)(25,26)(27,30)(28,32)(29,39)(31,34)(33,40)(35,38)(36,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,29,31,32)(26,28,34,39)(27,36,33,38)(30,35,40,37);(1,4)(2,3)(5,8)(6,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,30)(26,33)(27,34)(28,38)(29,35)(31,40)(32,37)(36,39)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971828);