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 | #40593 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4628 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)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(25,26)(27,32)(28,35)(29,30)(31,36)(33,38)(34,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27,31,37)(26,32,36,40)(28,29,38,34)(30,33,39,35);(1,3)(2,5)(4,8)(6,7)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,28)(26,33)(27,34)(29,37)(30,32)(31,38)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,29,31,34)(26,30,36,39)(27,28,37,38)(32,35,40,33);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30)(26,34)(27,33)(28,32)(29,36)(31,39)(35,37)(38,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,28)(26,35)(27,29)(30,32)(31,38)(33,36)(34,37)(39,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329023);