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 | |
4286 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)(11,20)(12,14)(13,17)(15,19)(16,21)(18,24)(22,23)(26,30)(27,35)(28,38)(34,40);(1,2)(3,4)(5,7)(6,8)(11,21)(13,22)(16,20)(17,23)(25,26,31,30)(27,36,35,32)(28,33,38,39)(29,40,37,34);(1,3)(2,6)(4,7)(5,8)(9,11,15,21)(10,16,19,20)(12,13,18,22)(14,23,24,17)(25,27,31,35)(26,32,30,36)(28,37,38,29)(33,40,39,34);(3,8)(4,6)(9,12,15,18)(10,14,19,24)(11,22,21,13)(16,17,20,23)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(1,4)(2,3)(5,6)(7,8)(9,13,15,22)(10,17,19,23)(11,12,21,18)(14,16,24,20)(25,29,31,37)(26,34,30,40)(27,28,35,38)(32,33,36,39);(1,2)(3,4)(5,7)(6,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,24)(20,22)(21,23)(25,30,31,26)(27,36,35,32)(28,39,38,33)(29,40,37,34)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8877282);