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 | |
4678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(10,15)(11,18)(17,21)(19,24)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(2,5)(3,8)(9,10,14,15)(11,16,18,12)(13,21,22,17)(19,20,24,23)(26,31)(28,32)(29,38)(34,40);(1,3,6,8)(2,7,5,4)(11,18)(12,16)(19,24)(20,23)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(1,5,6,2)(3,4,8,7)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669269);