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 | |
4626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,18,23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,2)(3,7)(4,5)(6,8)(10,15)(13,23)(16,19)(17,21)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(2,5)(6,7)(11,20)(12,22)(16,19)(17,21)(26,30)(27,34)(32,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,19,23,16)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,21)(12,19)(14,23)(15,24)(16,22)(17,20)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=gh,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, 8667918);