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 | #2675 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(2,4)(5,7)(11,19)(13,23)(15,24)(17,20)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,4)(3,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3,6,8)(2,7,4,5)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,4,6,2)(3,5,8,7)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24);(1,5)(2,3)(4,8)(6,7)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148156);