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 | #55643 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,14)(13,17)(15,19)(16,21)(18,24)(22,23)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,2)(3,6)(4,5)(7,8)(11,21)(13,22)(16,20)(17,23)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,5)(3,7)(9,11,15,21)(10,16,19,20)(12,13,18,22)(14,23,24,17)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(3,7)(6,8)(9,12,15,18)(10,14,19,24)(11,22,21,13)(16,17,20,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(9,13,15,22)(10,17,19,23)(11,12,21,18)(14,16,24,20);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,24)(20,22)(21,23)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9422481);