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 |
| prime radical | #57981 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,18)(12,21)(13,20)(14,16)(15,22)(17,23)(19,24)(25,26)(27,32)(28,34)(29,35)(30,40)(31,33)(36,37)(38,39);(2,6)(3,7)(9,11,14,22)(10,15,16,18)(12,20,23,19)(13,21,24,17)(25,27,31,30)(26,32,33,40)(28,36,35,38)(29,39,34,37);(1,3)(2,4)(5,7)(6,8)(10,16)(12,23)(15,18)(19,20)(26,33)(28,35)(32,40)(36,38);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,20)(13,15)(14,23)(16,21)(18,24)(19,22)(25,28,31,35)(26,34,33,29)(27,36,30,38)(32,37,40,39);(9,11)(10,18)(12,19)(13,17)(14,22)(15,16)(20,23)(21,24)(25,29,31,34)(26,35,33,28)(27,37,30,39)(32,36,40,38);(9,13)(10,19)(11,21)(12,15)(14,24)(16,20)(17,22)(18,23)(25,30,31,27)(26,32,33,40)(28,38,35,36)(29,39,34,37)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123693);