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 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,4,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(1,3,6,4)(2,7,8,5)(11,17)(15,21)(18,23)(22,24)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(10,14)(15,21)(16,19)(22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(2,8)(3,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,4,6,3)(2,7,8,5)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,27)(26,28)(29,37)(30,35)(31,33)(32,39)(34,38)(36,40);(1,5,6,7)(2,4,8,3)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154576);