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 | #53408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,4)(9,10,15,18)(11,19,23,16)(12,13,17,14)(20,22,24,21)(25,26,31,34)(27,30,33,29)(28,36,38,32)(35,39,40,37);(1,2,5,6)(3,8,4,7)(9,11,15,23)(10,16,18,19)(12,20,17,24)(13,21,14,22)(25,27,31,33)(26,29,34,30)(28,35,38,40)(32,39,36,37);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,20)(14,18)(15,17)(16,21)(19,22)(23,24)(25,28)(26,32)(27,35)(29,39)(30,37)(31,38)(33,40)(34,36);(1,3,5,4)(2,7,6,8)(9,13,15,14)(10,12,18,17)(11,21,23,22)(16,20,19,24)(25,29,31,30)(26,33,34,27)(28,37,38,39)(32,35,36,40);(1,4,5,3)(2,7,6,8)(9,13,15,14)(10,17,18,12)(11,22,23,21)(16,20,19,24)(25,30,31,29)(26,27,34,33)(28,37,38,39)(32,35,36,40);(9,14,15,13)(10,12,18,17)(11,22,23,21)(16,20,19,24)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=g,f^2=gh,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, 8010425);