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 | |
4552 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)(5,6)(7,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,40)(38,39);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27,30,35)(26,31,34,40)(28,29,36,39)(32,38,37,33);(1,3,5,7)(2,4,6,8)(10,11)(15,17)(16,18)(23,24)(27,35)(28,36)(31,40)(32,37);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,28)(26,32)(27,29)(30,36)(31,38)(33,40)(34,37)(35,39);(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,40)(36,37);(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,29)(26,33)(27,36)(28,35)(30,39)(31,32)(34,38)(37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641496);