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 | |
4644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,23)(19,21)(25,26)(27,34)(28,36)(29,33)(30,31)(32,37)(35,39)(38,40);(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,16,18,13)(12,17,20,23)(19,24,21,22)(25,27,31,33)(26,29,30,34)(28,35,37,40)(32,39,36,38);(1,3,5,8)(2,6,4,7)(11,18)(12,20)(13,16)(17,23)(26,30)(28,37)(29,34)(35,40);(3,8)(6,7)(9,11)(10,16)(12,21)(13,15)(14,18)(17,22)(19,20)(23,24)(25,28,31,37)(26,32,30,36)(27,35,33,40)(29,39,34,38);(1,3)(2,7)(4,6)(5,8)(9,12,14,20)(10,17,15,23)(11,19,18,21)(13,22,16,24)(25,29,31,34)(26,33,30,27)(28,38,37,39)(32,35,36,40);(1,4,5,2)(3,6,8,7)(9,13,14,16)(10,11,15,18)(12,22,20,24)(17,19,23,21)(25,30,31,26)(27,34,33,29)(28,36,37,32)(35,38,40,39)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=h,e^2=g,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, 8935182);