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 | #40044 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,33)(27,32,36,39)(28,34,38,30)(29,37,40,35);(3,7)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27)(26,32)(28,37)(29,30)(31,36)(33,39)(34,40)(35,38);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(26,33)(27,36)(30,34)(35,37);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,34)(27,37)(29,39)(30,33)(31,38)(32,40)(35,36);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29)(26,35)(27,34)(28,32)(30,36)(31,40)(33,37)(38,39);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,34)(26,28,33,38)(27,29,36,40)(32,37,39,35)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306613);