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 | |
4606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,18)(13,22)(14,20)(15,17)(19,21)(23,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(3,7)(6,8)(9,11)(10,16)(12,21)(13,23)(14,17)(15,20)(18,19)(22,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(13,21)(18,24)(25,28,30,33)(26,27,32,35)(29,36,38,39)(31,34,37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,13)(14,19)(15,23)(16,22)(17,24)(20,21)(25,26)(27,28)(29,37)(30,32)(31,38)(33,35)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,12)(14,18)(15,21)(16,24)(17,22)(20,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,14)(10,20)(11,17)(12,22)(13,18)(15,16)(19,23)(21,24)(25,29)(26,31)(27,34)(28,36)(30,38)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^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, 8490332);