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 | |
4449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,24)(21,23)(25,26,31,35)(27,28,36,32)(29,33,37,40)(30,34,39,38);(3,7)(6,8)(9,11,15,19)(10,16,14,22)(12,20,21,13)(17,24,23,18)(25,27)(26,32)(28,35)(29,30)(31,36)(33,38)(34,40)(37,39);(1,2,5,4)(3,6,7,8)(10,14)(11,19)(13,20)(17,23)(29,37)(30,39)(33,40)(34,38);(9,12)(10,17)(11,20)(13,19)(14,23)(15,21)(16,24)(18,22)(25,28,31,32)(26,27,35,36)(29,38,37,34)(30,33,39,40);(1,3,5,7)(2,6,4,8)(9,13,15,20)(10,18,14,24)(11,21,19,12)(16,23,22,17)(25,29)(26,33)(27,30)(28,34)(31,37)(32,38)(35,40)(36,39);(1,4,5,2)(3,8,7,6)(9,14,15,10)(11,22,19,16)(12,23,21,17)(13,24,20,18)(25,30)(26,34)(27,29)(28,33)(31,39)(32,40)(35,38)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853391);