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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4575 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)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,2,5,4)(3,6,7,8)(10,14)(15,17)(16,20)(22,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(2,4)(3,7)(10,14)(11,18)(12,21)(22,23)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(1,3,5,7)(2,6,4,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28)(26,27)(29,37)(30,34)(31,33)(32,39)(35,40)(36,38);(1,3)(2,6)(4,8)(5,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8939652);