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 | |
4230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,20)(13,18)(14,15)(16,22)(19,21)(23,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,40)(36,39);(3,7)(5,8)(10,15)(13,17)(16,20)(21,23)(26,31)(28,33)(32,34)(35,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(16,20)(19,24)(25,26,30,31)(27,32,37,34)(28,29,33,38)(35,36,39,40);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,27,30,37)(26,32,31,34)(28,35,33,39)(29,36,38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,22)(10,16,15,20)(11,19,18,24)(13,21,17,23)(25,28,30,33)(26,29,31,38)(27,35,37,39)(32,36,34,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,24)(22,23)(25,29)(26,33)(27,36)(28,31)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=i,e^2=g,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, 8662040);