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 | |
4305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26)(27,32)(28,29)(30,38)(31,33)(34,39)(35,37)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27,31,36)(26,32,33,40)(28,37,39,30)(29,35,34,38);(1,3,5,8)(2,7,6,4)(10,17)(11,20)(12,22)(18,24)(26,33)(27,36)(28,39)(35,38);(1,4)(2,8)(3,6)(5,7)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,28)(26,34)(27,37)(29,33)(30,36)(31,39)(32,38)(35,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,29,31,34)(26,28,33,39)(27,38,36,35)(30,40,37,32);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,30,31,37)(26,35,33,38)(27,28,36,39)(29,32,34,40)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,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, 9637535);