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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,13,16,23)(12,19,22,15)(18,24,20,21)(26,29)(27,35)(28,36)(34,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,13,17,23)(12,20,22,18)(15,24,19,21)(25,26)(27,32)(28,37)(29,31)(30,34)(33,36)(35,38)(39,40);(2,3)(6,7)(9,12,14,22)(10,15,17,19)(11,18,16,20)(13,21,23,24)(25,27,31,35)(26,32,29,38)(28,30,36,40)(33,39,37,34);(4,8)(6,7)(9,11)(10,13)(12,18)(14,16)(15,21)(17,23)(19,24)(20,22)(25,28)(26,33)(27,30)(29,37)(31,36)(32,39)(34,38)(35,40);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,19)(23,24)(25,29,31,26)(27,32,35,38)(28,37,36,33)(30,39,40,34);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,17)(12,21)(14,23)(15,20)(18,19)(22,24)(25,30)(26,34)(27,36)(28,35)(29,39)(31,40)(32,33)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520185);