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 | |
4704 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)(3,5)(4,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(27,33)(31,34)(35,39)(38,40);(2,7)(3,8)(10,15)(12,21)(16,22)(18,19)(25,26,29,30)(27,31,33,34)(28,32,36,37)(35,38,39,40);(1,3,6,8)(2,4,7,5)(12,21)(13,23)(17,24)(18,19)(26,30)(28,36)(31,34)(35,39);(1,4,6,5)(2,3,7,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,22)(19,21)(23,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,24)(16,23)(18,20)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,5,6,4)(2,8,7,3)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,22,21,16)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946447);