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 | |
4260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,18,16)(13,17,22,23)(19,20,21,24)(25,26)(27,32)(28,29)(30,40)(31,34)(33,38)(35,39)(36,37);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,23)(19,21)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,2,4,6)(3,5,7,8)(10,15)(12,16)(17,23)(20,24)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(3,7)(5,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,22)(23,24)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,18)(13,20,22,24)(17,19,23,21)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8433939);