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 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,23)(14,15)(16,21)(18,24)(20,22)(25,26)(27,35)(28,33)(29,31)(30,34)(32,36)(37,38)(39,40);(1,2,5,4)(3,7,8,6)(10,15)(12,22)(16,19)(18,23)(25,27)(26,32)(28,30)(29,35)(31,36)(33,39)(34,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11,14,21)(10,16,15,19)(12,23,22,18)(13,17,24,20)(25,28,31,37)(26,33,29,38)(27,30,36,40)(32,39,35,34);(3,8)(6,7)(9,11)(10,16)(12,23)(13,20)(14,21)(15,19)(17,24)(18,22)(25,29,31,26)(27,35,36,32)(28,33,37,38)(30,39,40,34);(1,3)(2,7)(4,6)(5,8)(9,12,14,22)(10,17,15,20)(11,18,21,23)(13,19,24,16)(25,28)(26,33)(27,30)(29,38)(31,37)(32,39)(34,35)(36,40);(1,4,5,2)(3,6,8,7)(9,13)(10,18)(11,20)(12,19)(14,24)(15,23)(16,22)(17,21)(25,30)(26,34)(27,28)(29,39)(31,40)(32,38)(33,35)(36,37)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935756);