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 | |
4299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26)(27,30)(28,32)(29,31)(33,34)(35,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,40)(37,39);(1,2,4,6)(3,7,8,5)(10,17)(11,12)(18,24)(19,20)(26,31)(27,34)(28,36)(38,39);(3,8)(5,7)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,28,29,36)(26,32,31,37)(27,35,34,40)(30,38,33,39);(1,3)(2,5)(4,8)(6,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,26)(27,33)(28,37)(29,31)(30,34)(32,36)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,27)(26,33)(28,35)(29,34)(30,31)(32,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9397793);