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 | #57218 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4458 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)(11,19)(12,13)(14,17)(15,18)(16,23)(20,22)(21,24)(25,26,30,29)(27,31,35,34)(28,32,37,38)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,30,35)(26,31,29,34)(28,36,37,33)(32,40,38,39);(1,3,5,7)(2,6,4,8)(9,12,15,17)(10,13,18,14)(11,20,23,24)(16,21,19,22)(25,28,30,37)(26,32,29,38)(27,33,35,36)(31,39,34,40);(1,2,5,4)(3,8,7,6)(9,13,15,14)(10,17,18,12)(11,21,23,22)(16,24,19,20)(25,29,30,26)(27,34,35,31)(28,32,37,38)(33,39,36,40);(1,4,5,2)(3,8,7,6)(9,13,15,14)(10,12,18,17)(11,22,23,21)(16,24,19,20)(27,35)(31,34)(33,36)(39,40);(9,14,15,13)(10,17,18,12)(11,22,23,21)(16,24,19,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=gi,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, 8012361);