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 | #38564 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4607 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)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26)(27,35)(28,37)(29,36)(30,34)(31,32)(33,39)(38,40);(2,5)(6,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,27,31,36)(26,29,32,35)(28,38,39,30)(33,40,37,34);(1,2,4,5)(3,6,7,8)(13,22)(16,20)(18,21)(23,24)(26,32)(29,35)(33,37)(34,40);(3,7)(6,8)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,28,31,39)(26,33,32,37)(27,30,36,38)(29,34,35,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,29)(26,27)(28,34)(30,33)(31,35)(32,36)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,15)(13,18)(14,19)(16,24)(20,23)(21,22)(25,30)(26,34)(27,28)(29,33)(31,38)(32,40)(35,37)(36,39)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^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, 8506750);