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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,22)(18,21)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(2,5)(6,8)(10,15)(13,24)(16,19)(17,22)(26,31)(27,33)(32,37)(34,40);(3,7)(6,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,17,24,22)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180660);