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 | #2540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4258 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)(11,18)(16,22)(19,24)(20,23)(25,26)(27,36)(28,38)(29,37)(30,33)(31,32)(34,39)(35,40);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(26,32)(27,30)(34,38)(35,37);(10,15)(12,21)(16,22)(19,24)(25,27,31,30)(26,33,32,36)(28,35,39,37)(29,34,40,38);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24)(25,28,31,39)(26,34,32,38)(27,37,30,35)(29,36,40,33);(9,12,14,21)(10,17,15,13)(11,19,18,24)(16,23,22,20)(25,29,31,40)(26,35,32,37)(27,38,30,34)(28,36,39,33);(9,13,14,17)(10,12,15,21)(11,20,18,23)(16,19,22,24)(25,30,31,27)(26,33,32,36)(28,35,39,37)(29,38,40,34)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895164);