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 | |
4241 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)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26)(27,34)(28,37)(29,32)(30,38)(31,33)(35,36)(39,40);(2,5)(6,7)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,22,24,19)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(19,22)(20,21)(26,33)(28,38)(32,34)(35,39);(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,19,24,22)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(1,2)(3,7)(4,5)(6,8)(9,13)(10,19)(11,21)(12,20)(14,16)(15,23)(17,22)(18,24)(25,29,31,27)(26,32,33,34)(28,39,38,35)(30,36,37,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,24)(10,20,17,21)(11,22,12,19)(13,18,23,16)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9688406);