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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,13,17)(11,15,19,14)(12,20,22,16)(18,23,21,24)(25,26)(27,31)(28,33)(29,39)(30,35)(32,38)(34,37)(36,40);(3,7)(6,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,23)(20,22)(21,24)(25,27,30,36)(26,31,35,40)(28,37,38,29)(32,39,33,34);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,17,15)(12,21,22,18)(16,24,20,23)(25,28,30,38)(26,32,35,33)(27,29,36,37)(31,34,40,39);(9,10)(11,14)(12,16)(13,17)(15,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,34)(30,35)(33,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,23)(21,22)(25,28)(26,33)(27,29)(30,38)(31,39)(32,35)(34,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12,13,22)(10,16,17,20)(11,18,19,21)(14,23,15,24)(25,29)(26,34)(27,28)(30,37)(31,32)(33,40)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851735);