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 | |
4600 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)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26,31,32)(27,33,28,37)(29,34,40,30)(35,36,38,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(26,32)(29,40)(33,37)(35,38);(2,5)(6,7)(10,16)(13,24)(15,19)(17,21)(25,27,31,28)(26,33,32,37)(29,35,40,38)(30,39,34,36);(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,28,31,27)(26,33,32,37)(29,38,40,35)(30,39,34,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,29)(26,34)(27,35)(28,38)(30,32)(31,40)(33,36)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,30,31,34)(26,29,32,40)(27,36,28,39)(33,38,37,35)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8105118);