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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(27,33)(30,34)(35,39)(38,40);(3,7)(5,8)(10,14)(15,17)(16,20)(23,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,2)(3,5)(4,6)(7,8)(11,18)(15,17)(19,21)(23,24)(25,26,29,32)(27,30,33,34)(28,31,37,36)(35,38,39,40);(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,37)(38,39);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28,29,37)(26,31,32,36)(27,35,33,39)(30,38,34,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8795285);