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 | #44165 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18);(2,6)(3,7)(10,15)(12,22)(16,19)(18,23)(25,26,31,29)(27,34,28,32)(30,33,38,37)(35,40,36,39);(1,2)(3,4)(5,6)(7,8)(10,15)(11,20)(17,21)(18,23)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(2,6)(3,7)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,7)(2,4,6,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,16,24,19)(25,29,31,26)(27,34,28,32)(30,33,38,37)(35,39,36,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=hi,e^2=gi,f^2=g,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, 10088270);