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 | |
4289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(7,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,30)(28,29)(31,32)(33,37)(34,40)(35,38)(36,39);(1,2,6,5)(3,7,4,8)(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(25,27,31,34)(26,30,32,40)(28,35,37,39)(29,38,33,36);(1,2,6,5)(3,8,4,7)(26,32)(27,34)(28,37)(36,38);(1,3,6,4)(2,7,5,8)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,28)(26,33)(27,35)(29,32)(30,36)(31,37)(34,39)(38,40);(1,4,6,3)(2,7,5,8)(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24)(25,29,31,33)(26,28,32,37)(27,36,34,38)(30,39,40,35);(1,5,6,2)(3,8,4,7)(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)(25,30)(26,34)(27,32)(28,36)(29,35)(31,40)(33,39)(37,38)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9637845);