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 | #35568 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,13)(14,23)(15,17)(18,22)(19,20)(21,24)(26,31)(29,33)(32,38)(35,39);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,17,24)(12,20,22,14)(13,19,18,23)(25,26,30,31)(27,32,36,38)(28,33,37,29)(34,39,40,35);(10,17)(13,18)(16,24)(19,23)(26,31)(28,37)(32,38)(34,40);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,20)(13,17)(14,21)(15,22)(16,23)(19,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,33,39);(9,13,15,18)(10,12,17,22)(11,19,21,23)(14,16,20,24)(25,28)(26,33)(27,34)(29,31)(30,37)(32,39)(35,38)(36,40);(9,14,15,20)(10,19,17,23)(11,12,21,22)(13,16,18,24)(25,29,30,33)(26,28,31,37)(27,35,36,39)(32,34,38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550651);