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 | |
4414 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)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,14,17)(13,20,21,19)(18,24,22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(10,14)(11,12)(13,21)(23,24)(27,32)(28,31)(29,37)(33,40);(3,8)(6,7)(9,12,15,11)(10,17,14,16)(13,19,21,20)(18,23,22,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,38)(36,40)(37,39);(1,3)(2,7)(4,6)(5,8)(9,13,15,21)(10,18,14,22)(11,19,12,20)(16,23,17,24)(25,28,30,31)(26,32,34,27)(29,36,37,38)(33,39,40,35);(1,4,5,2)(3,6,8,7)(9,14,15,10)(11,17,12,16)(13,22,21,18)(19,24,20,23)(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=hi,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, 8936049);