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 | |
4666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,14)(13,20)(15,19)(17,22)(18,23)(21,24)(25,26,30,34)(27,33,36,29)(28,31,38,32)(35,39,40,37);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,15,12,16)(13,18,22,21)(17,23,20,24)(25,27)(26,29)(28,35)(30,36)(31,37)(32,39)(33,34)(38,40);(1,3)(2,4)(5,8)(6,7)(10,12)(15,16)(17,20)(23,24)(25,28,30,38)(26,31,34,32)(27,35,36,40)(29,37,33,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,10)(11,16,19,15)(13,20,22,17)(18,24,21,23)(25,28)(26,32)(27,35)(29,39)(30,38)(31,34)(33,37)(36,40);(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,29)(26,27)(28,37)(30,33)(31,35)(32,40)(34,36)(38,39);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,27)(26,33)(28,35)(29,34)(30,36)(31,39)(32,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050546);