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 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26)(27,33)(28,38)(29,31)(30,34)(32,35)(36,40)(37,39);(2,5)(3,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27,32,29)(26,33,35,31)(28,37,30,36)(34,40,38,39);(1,3,6,8)(2,7,5,4)(11,17)(12,20)(15,19)(16,21)(25,28,32,30)(26,34,35,38)(27,36,29,37)(31,40,33,39);(1,4)(2,3)(5,8)(6,7)(9,12,14,20)(10,16,13,21)(11,18,17,23)(15,22,19,24)(25,29,32,27)(26,31,35,33)(28,36,30,37)(34,39,38,40);(1,5,6,2)(3,4,8,7)(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,30,32,28)(26,34,35,38)(27,37,29,36)(31,40,33,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,31)(26,27)(28,39)(29,35)(30,40)(32,33)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8829725);