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 | #48599 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,16,19,23)(12,17,21,20)(13,22,24,18)(25,26,31,32)(27,35,36,33)(28,39,40,29)(30,37,38,34);(1,2)(3,4)(5,7)(6,8)(9,10,14,15)(11,16,19,23)(12,20,21,17)(13,18,24,22)(26,32)(27,36)(29,39)(34,37);(1,3)(2,6)(4,7)(5,8)(10,15)(11,19)(12,21)(13,24)(25,27,31,36)(26,33,32,35)(28,37,40,34)(29,38,39,30);(3,8)(4,6)(9,11)(10,16)(12,22)(13,20)(14,19)(15,23)(17,24)(18,21)(25,28)(26,29)(27,37)(30,35)(31,40)(32,39)(33,38)(34,36);(1,4)(2,3)(5,6)(7,8)(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,23,24,16)(25,29)(26,28)(27,38)(30,36)(31,39)(32,40)(33,37)(34,35);(1,2)(3,4)(5,7)(6,8)(9,13)(10,18)(11,20)(12,23)(14,24)(15,22)(16,21)(17,19)(25,30,31,38)(26,34,32,37)(27,29,36,39)(28,35,40,33)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8876718);