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 | #53424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4323 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,19,21,16)(12,22,23,17)(13,18,24,20)(25,26,31,36)(27,32,38,40)(28,29,34,33)(30,39,37,35);(2,6)(3,7)(10,15)(12,23)(16,19)(18,20)(25,27,31,38)(26,32,36,40)(28,37,34,30)(29,35,33,39);(1,3,5,7)(2,4,6,8)(25,28)(26,33)(27,37)(29,36)(30,38)(31,34)(32,39)(35,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,18)(13,22)(14,21)(15,19)(17,24)(20,23)(25,28)(26,29)(27,37)(30,38)(31,34)(32,35)(33,36)(39,40);(9,12,14,23)(10,17,15,22)(11,20,21,18)(13,19,24,16)(25,29)(26,34)(27,35)(28,36)(30,32)(31,33)(37,40)(38,39);(9,13)(10,18)(11,17)(12,19)(14,24)(15,20)(16,23)(21,22)(25,30,31,37)(26,35,36,39)(27,28,38,34)(29,32,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10347234);