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 | |
4851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,16,12,19)(14,17,22,21)(18,23,20,24)(25,26)(27,36)(28,34)(29,33)(30,38)(31,37)(32,39)(35,40);(2,5)(6,8)(10,13)(16,19)(17,21)(23,24)(25,27)(26,32)(28,38)(29,35)(30,33)(31,39)(34,40)(36,37);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,13,19)(14,18,22,20)(17,23,21,24)(25,28,31,29)(26,33,37,34)(27,38,39,35)(30,36,40,32);(3,7)(6,8)(9,12,15,11)(10,16,13,19)(14,18,22,20)(17,24,21,23)(25,29,31,28)(26,34,37,33)(27,38,39,35)(30,36,40,32);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,16,12,19)(14,21,22,17)(18,23,20,24)(25,30)(26,35)(27,34)(28,32)(29,36)(31,40)(33,39)(37,38);(1,3)(2,6)(4,7)(5,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,27)(26,36)(28,38)(29,35)(30,34)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=g,e^2=gh,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, 8506700);