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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,31,34)(27,29,32,28)(30,38,39,33)(35,36,40,37);(1,2,5,4)(3,7,8,6)(10,15)(12,16)(17,22)(20,23)(25,27,31,32)(26,28,34,29)(30,35,39,40)(33,36,38,37);(2,4)(6,7)(13,21)(17,22)(18,24)(20,23)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,40,38,35);(3,8)(6,7)(9,11,14,19)(10,16,15,12)(13,18,21,24)(17,23,22,20)(25,29,31,28)(26,27,34,32)(30,37,39,36)(33,35,38,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8282014);