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 | #51061 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,6)(3,5,8,4)(9,10,15,20)(11,16,22,13)(12,18,24,17)(14,23,21,19)(25,26,31,28)(27,29,32,34)(30,33,39,36)(35,37,40,38);(1,3)(2,4)(5,6)(7,8)(9,11,15,22)(10,16,20,13)(12,21,24,14)(17,23,18,19)(27,32)(29,34)(35,40)(37,38);(9,12,15,24)(10,17,20,18)(11,14,22,21)(13,23,16,19)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(1,4,7,5)(2,8,6,3)(9,12)(10,18)(11,21)(13,23)(14,22)(15,24)(16,19)(17,20)(25,28,31,26)(27,34,32,29)(30,36,39,33)(35,38,40,37);(1,5,7,4)(2,8,6,3)(9,13)(10,11)(12,23)(14,17)(15,16)(18,21)(19,24)(20,22)(25,29,31,34)(26,32,28,27)(30,38,39,37)(33,35,36,40);(1,6,7,2)(3,5,8,4)(9,14,15,21)(10,19,20,23)(11,12,22,24)(13,18,16,17)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770602);