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 | #55593 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(3,7)(9,10,15,12)(11,19,21,16)(13,17,14,18)(20,24,22,23)(25,26,30,34)(27,31,32,28)(29,33,39,40)(35,38,37,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23)(25,27)(26,31)(28,34)(29,37)(30,32)(33,36)(35,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,10)(11,19,21,16)(13,18,14,17)(20,24,22,23)(25,28,30,31)(26,32,34,27)(29,38,39,36)(33,35,40,37);(1,2,5,4)(3,8,7,6)(10,12)(16,19)(17,18)(23,24)(25,29,30,39)(26,33,34,40)(27,35,32,37)(28,36,31,38);(1,4,5,2)(3,8,7,6)(9,13,15,14)(10,17,12,18)(11,20,21,22)(16,23,19,24)(26,34)(27,32)(33,40)(35,37);(9,14,15,13)(10,18,12,17)(11,20,21,22)(16,23,19,24)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=h,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, 8012825);