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 |
| prime radical | #34962 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 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)(9,10,15,17)(11,16,20,23)(12,18,22,14)(13,21,24,19)(25,26)(27,36)(28,30)(29,38)(31,35)(32,39)(33,40)(34,37);(3,7)(6,8)(9,11)(10,16)(12,21)(13,14)(15,20)(17,23)(18,24)(19,22)(25,27,31,39)(26,32,35,36)(28,37,40,29)(30,38,33,34);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(14,18)(19,21)(25,28,31,40)(26,33,35,30)(27,37,39,29)(32,34,36,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,21)(13,23)(14,17)(15,22)(16,24)(19,20)(25,29)(26,34)(27,28)(30,32)(31,37)(33,36)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,18)(12,16)(14,20)(15,24)(17,21)(22,23)(25,27)(26,32)(28,37)(29,40)(30,38)(31,39)(33,34)(35,36);(9,14,15,18)(10,12,17,22)(11,13,20,24)(16,21,23,19)(25,30,31,33)(26,28,35,40)(27,38,39,34)(29,32,37,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306601);