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 | #40959 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4326 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)(11,17)(12,15)(13,21)(14,16)(18,22)(19,23)(20,24)(25,26,30,34)(27,31,32,28)(29,33,39,40)(35,37,38,36);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,37)(36,40)(38,39);(2,6)(3,7)(10,16)(15,17)(18,21)(23,24)(25,28,30,31)(26,32,34,27)(29,36,39,37)(33,38,40,35);(3,7)(4,8)(9,12,14,11)(10,17,16,15)(13,19,22,20)(18,23,21,24)(25,29)(26,33)(27,35)(28,36)(30,39)(31,37)(32,38)(34,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,23)(21,22)(25,28)(26,27)(29,37)(30,31)(32,34)(33,38)(35,40)(36,39);(1,4)(2,3)(5,8)(6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,27)(26,28)(29,38)(30,32)(31,34)(33,37)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8971131);