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 | #35907 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,23)(22,24)(25,26)(27,35)(28,36)(29,33)(30,38)(31,32)(34,37)(39,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,27,31,36)(26,28,32,35)(29,37,39,30)(33,38,40,34);(1,3,5,8)(2,6,7,4)(12,21)(15,20)(18,22)(23,24)(26,32)(28,35)(33,40)(34,38);(3,8)(4,6)(9,12,13,21)(10,15,17,20)(11,18,19,22)(14,23,16,24)(25,28,31,35)(26,27,32,36)(29,38,39,34)(30,33,37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,22)(13,19)(14,17)(15,23)(18,21)(20,24)(25,29,31,39)(26,33,32,40)(27,30,36,37)(28,34,35,38);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,23)(20,21)(22,24)(25,30)(26,34)(27,29)(28,33)(31,37)(32,38)(35,40)(36,39)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9044904);