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 | #2613 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,18,22)(12,15,20,19)(17,21,24,23)(26,30)(27,33)(28,36)(38,40);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8435322);