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 | #44206 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(4,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(26,31)(27,36)(28,35)(38,39);(1,2,6,5)(3,8,7,4)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,27,30,36)(26,32,31,33)(28,37,35,40)(29,38,34,39);(2,5)(3,7)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,27)(26,33)(28,37)(29,39)(30,36)(31,32)(34,38)(35,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,28)(26,34)(27,37)(29,31)(30,35)(32,39)(33,38)(36,40);(1,5,6,2)(3,8,7,4)(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,29,30,34)(26,35,31,28)(27,38,36,39)(32,40,33,37)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627542);