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 | |
4334 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)(4,5)(6,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26,31,33)(27,37,38,32)(28,29,35,34)(30,40,39,36);(3,8)(6,7)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27)(26,32)(28,39)(29,40)(30,35)(31,38)(33,37)(34,36);(1,3)(2,6)(4,7)(5,8)(10,15)(14,21)(16,19)(22,23)(26,33)(27,38)(28,35)(36,40);(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28,31,35)(26,34,33,29)(27,39,38,30)(32,36,37,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29,31,34)(26,35,33,28)(27,36,38,40)(30,32,39,37);(1,4,5,2)(3,6,8,7)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,30,31,39)(26,36,33,40)(27,35,38,28)(29,32,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9347763);