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 | #40381 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,35)(28,33)(29,37)(30,31)(32,36)(34,40)(38,39);(1,2,4,6)(3,5,8,7)(12,19)(16,22)(18,23)(20,24)(26,31)(28,39)(32,35)(34,37);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(16,22)(18,23)(26,31)(27,36)(33,38)(34,37);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,36)(26,32,31,35)(28,34,39,37)(29,38,40,33);(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29,30,40)(26,34,31,37)(27,38,36,33)(28,32,39,35)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123864);