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 | |
4718 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,17,18,16)(12,21,23,13)(19,24,22,20)(25,26)(27,33)(28,37)(29,38)(30,32)(31,36)(34,40)(35,39);(2,6)(3,7)(10,15)(13,21)(16,17)(20,24)(25,27,30,36)(26,31,32,33)(28,34,38,39)(29,40,37,35);(1,3,5,7)(2,4,6,8)(11,18)(16,17)(19,22)(20,24)(26,32)(28,38)(31,33)(34,39);(1,4)(2,7)(3,6)(5,8)(9,11,14,18)(10,16,15,17)(12,22,23,19)(13,20,21,24)(25,28)(26,29)(27,34)(30,38)(31,40)(32,37)(33,35)(36,39);(9,12)(10,13)(11,19)(14,23)(15,21)(16,24)(17,20)(18,22)(25,27)(26,31)(28,39)(29,35)(30,36)(32,33)(34,38)(37,40);(9,13)(10,12)(11,20)(14,21)(15,23)(16,22)(17,19)(18,24)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757782);