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 | |
4607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,21)(22,24);(1,2,6,5)(3,8,4,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,3,6,4)(2,7,5,8)(13,21)(16,20)(18,24)(22,23)(26,31)(28,38)(32,40)(34,35);(1,4,6,3)(2,8,5,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,32)(28,35)(29,33)(30,37)(31,40)(34,38)(36,39);(1,5,6,2)(3,8,4,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,11)(10,17)(12,15)(13,18)(14,19)(16,23)(20,22)(21,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424685);