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 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(10,15)(11,16)(17,21)(18,23)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3)(2,8)(4,6)(5,7)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(27,33)(29,37)(31,34)(32,36);(1,2,7,6)(3,8,5,4)(11,16)(12,19)(18,23)(20,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,4)(2,5)(3,6)(7,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(1,5,7,3)(2,8,6,4)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,28,30,26)(27,31,33,34)(29,36,37,32)(35,38,39,40);(1,6,7,2)(3,8,5,4)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714071);