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 | #36738 | C2 |
| 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)(10,16)(11,12)(13,23)(19,22)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(2,5)(6,7)(9,10,15,16)(11,17,12,20)(13,24,23,14)(18,19,21,22)(27,35)(28,38)(31,40)(32,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,20)(13,18,23,21)(14,22,24,19)(25,27)(26,31)(28,36)(29,32)(30,35)(33,38)(34,40)(37,39);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,17,16,20)(13,21,23,18)(14,22,24,19);(9,13)(10,14)(11,18)(12,21)(15,23)(16,24)(17,22)(19,20)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(9,14)(10,13)(11,19)(12,22)(15,24)(16,23)(17,21)(18,20)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109178);