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 | |
4581 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)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26)(27,33)(28,29)(30,35)(31,32)(34,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,13,18)(12,21,22,19)(17,24,20,23)(26,32)(29,34)(33,35)(37,39);(2,3)(6,7)(10,13)(16,18)(17,20)(23,24)(25,27,31,30)(26,33,32,35)(28,36,38,40)(29,39,34,37);(4,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24)(25,28)(26,34)(27,36)(29,32)(30,40)(31,38)(33,37)(35,39);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,16,14,18)(12,20,22,17)(19,23,21,24)(25,29,31,34)(26,28,32,38)(27,37,30,39)(33,40,35,36);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,13,18)(12,19,22,21)(17,24,20,23)(25,30,31,27)(26,35,32,33)(28,36,38,40)(29,39,34,37)Polycyclic presentation:
2^9|b^2=h,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8519416);