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 | #46396 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,19)(13,16)(14,15)(18,24)(20,21)(22,23)(25,26)(27,31)(28,38)(29,30)(32,33)(34,39)(35,36)(37,40);(2,5)(6,8)(9,11,14,19)(10,12,15,17)(13,18,22,21)(16,20,23,24)(25,27,32,37)(26,31,33,40)(28,35,29,34)(30,39,38,36);(1,2,4,5)(3,6,7,8)(10,15)(12,17)(16,23)(20,24)(25,28,32,29)(26,30,33,38)(27,34,37,35)(31,36,40,39);(3,7)(6,8)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,29,32,28)(26,30,33,38)(27,35,37,34)(31,36,40,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,30)(26,28)(27,36)(29,33)(31,34)(32,38)(35,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,31,32,40)(26,27,33,37)(28,36,29,39)(30,34,38,35)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8508095);