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 | |
4236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(26,31)(29,37)(32,38)(36,40);(2,6)(3,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(28,35)(32,38)(34,39)(36,40);(1,3)(2,4)(5,7)(6,8)(10,14)(15,17)(16,20)(22,23)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662518);