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 | |
4447 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(26,30)(27,33)(32,35)(34,36);(1,2)(3,7)(4,6)(5,8)(11,17)(15,21)(18,23)(22,24)(25,26,29,30)(27,31,33,38)(28,35,37,32)(34,40,36,39);(1,2,4,6)(3,5,8,7)(10,14)(15,21)(16,19)(22,24)(25,27,29,33)(26,31,30,38)(28,36,37,34)(32,40,35,39);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,31)(28,34)(29,33)(30,38)(32,39)(35,40)(36,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,29,37)(26,32,30,35)(27,34,33,36)(31,39,38,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)Polycyclic presentation:
2^9|b^2=g,c^2=gi,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=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9152368);