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 | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(26,27)(31,33)(32,34)(38,39);(2,6)(4,7)(10,16)(11,17)(18,21)(19,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,7)(2,3,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=h,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102548);