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 | |
4420 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)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(25,26,31,33)(27,32,29,30)(28,34,38,40)(35,37,36,39);(3,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,27)(26,32)(28,36)(29,31)(30,33)(34,39)(35,38)(37,40);(1,2,5,4)(3,6,7,8)(11,18)(14,17)(19,24)(22,23)(26,33)(27,29)(34,40)(35,36);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,28)(26,34)(27,35)(29,36)(30,39)(31,38)(32,37)(33,40);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,29,31,27)(26,32,33,30)(28,36,38,35)(34,37,40,39);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30,31,32)(26,27,33,29)(28,37,38,39)(34,36,40,35)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=hi,e^2=h,f^2=gi,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, 8556184);