Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #53659 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4477 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)(3,6)(4,8)(5,7)(9,10,15,12)(11,19,22,16)(13,18,14,17)(20,23,21,24)(25,26,31,34)(27,30,33,29)(28,36,38,32)(35,39,40,37);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,27,31,33)(26,29,34,30)(28,35,38,40)(32,39,36,37);(1,4,5,6)(2,3,7,8)(9,12,15,10)(11,19,22,16)(13,17,14,18)(20,23,21,24)(25,28)(26,32)(27,35)(29,39)(30,37)(31,38)(33,40)(34,36);(3,8)(4,6)(9,13,15,14)(10,17,12,18)(11,20,22,21)(16,23,19,24)(25,29,31,30)(26,33,34,27)(28,37,38,39)(32,35,36,40);(2,7)(3,8)(9,13,15,14)(10,18,12,17)(11,21,22,20)(16,23,19,24)(25,30,31,29)(26,27,34,33)(28,37,38,39)(32,35,36,40);(9,14,15,13)(10,17,12,18)(11,21,22,20)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012785);