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 | |
4578 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)(3,8)(4,5)(6,7)(9,10,15,18)(11,16,19,13)(12,14,17,23)(20,21,24,22)(26,31)(28,32)(33,39)(36,37);(1,3,6,5)(2,4,7,8)(9,11)(10,16)(12,20)(13,18)(14,21)(15,19)(17,24)(22,23)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(2,7)(4,8)(11,19)(13,16)(20,24)(21,22)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(2,7)(3,5)(9,12)(10,14)(11,20)(13,22)(15,17)(16,21)(18,23)(19,24)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,22)(14,20)(15,16)(17,21)(18,19)(23,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(1,5,6,3)(2,4,7,8)(9,14)(10,17)(11,21)(12,18)(13,20)(15,23)(16,24)(19,22)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8494653);