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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,35)(28,38)(29,40)(30,34)(31,37)(32,39)(33,36);(1,2)(3,6)(4,5)(7,8)(25,27,30,37)(26,31,34,35)(28,36,39,29)(32,33,38,40);(10,14)(15,17)(16,21)(22,23)(25,28,30,39)(26,32,34,38)(27,29,37,36)(31,40,35,33);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,32)(27,36)(29,37)(30,39)(31,33)(34,38)(35,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,30,36)(26,33,34,40)(27,28,37,39)(31,38,35,32);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,40)(30,34)(33,36)(35,37)(38,39)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7837704);