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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,26)(27,35)(28,37)(29,34)(30,38)(31,33)(32,36)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27,31,36)(26,32,33,35)(28,29,38,39)(30,34,37,40);(1,3,5,4)(2,6,7,8)(26,33)(27,36)(28,38)(34,40);(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28)(26,30)(27,29)(31,38)(32,34)(33,37)(35,40)(36,39);(1,4,5,3)(2,6,7,8)(9,13,15,10)(11,17,12,16)(14,18,22,21)(19,23,20,24)(25,29)(26,34)(27,28)(30,32)(31,39)(33,40)(35,37)(36,38);(1,2)(3,6)(4,8)(5,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30)(26,28)(27,34)(29,32)(31,37)(33,38)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613859);