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 | |
4470 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,18,23,24)(25,26)(27,30)(28,35)(29,31)(32,37)(33,39)(34,40)(36,38);(2,6)(3,7)(10,15)(13,23)(16,19)(17,21)(25,27,29,33)(26,30,31,39)(28,36,37,34)(32,40,35,38);(1,3,5,7)(2,4,6,8)(11,20)(12,22)(16,19)(17,21)(26,31)(27,33)(32,35)(34,36);(1,4)(2,7)(3,6)(5,8)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,26)(27,30)(28,32)(29,31)(33,39)(34,38)(35,37)(36,40);(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,19,23,16)(25,27)(26,30)(28,34)(29,33)(31,39)(32,38)(35,40)(36,37);(9,13)(10,18)(11,21)(12,19)(14,23)(15,24)(16,22)(17,20)(25,28,29,37)(26,32,31,35)(27,34,33,36)(30,38,39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=hi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714118);