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 | |
4403 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,22)(13,18)(14,15)(17,23)(19,21)(20,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,20)(13,17)(14,21)(15,19)(18,23)(22,24)(26,30)(31,33)(32,38)(39,40);(1,2)(3,5)(4,6)(7,8)(10,14)(11,19)(12,23)(13,24)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,17,14,22)(11,20,19,18)(13,21,24,16)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(9,13,15,24)(10,18,14,20)(11,17,19,22)(12,16,23,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(9,14,15,10)(11,21,19,16)(12,22,23,17)(13,20,24,18)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8965112);