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 |
| prime radical | #46504 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(1,2,5,4)(3,7,8,6)(10,15)(12,22)(16,19)(18,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(10,15)(11,20)(12,22)(13,24)(27,32)(28,31)(29,37)(33,40);(3,8)(6,7)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,27)(26,31)(28,34)(29,35)(30,32)(33,38)(36,40)(37,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,28,30,31)(26,32,34,27)(29,36,37,38)(33,39,40,35);(1,4,5,2)(3,6,8,7)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,29,30,37)(26,33,34,40)(27,35,32,39)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935793);