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 | |
4351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,24)(12,14,19,22)(13,21,23,18)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,20)(10,16,17,24)(12,18,19,21)(13,22,23,14)(26,28)(27,33)(32,36)(35,39);(2,5)(6,7)(10,17)(11,20)(14,22)(18,21)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(9,12,15,19)(10,14,17,22)(11,21,20,18)(13,16,23,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,14)(12,16)(15,23)(17,21)(19,24)(20,22)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,13)(12,17)(15,22)(16,21)(18,24)(20,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8105380);