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 | |
4661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,33)(28,36)(29,32)(30,34)(31,38)(35,39)(37,40);(2,3)(4,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26,29,32)(27,30,34,33)(28,31,38,36)(35,39,37,40);(1,2,5,3)(4,6,8,7)(10,14)(15,18)(16,20)(22,24)(27,34)(30,33)(35,37)(39,40);(2,3)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,30)(28,37)(29,34)(31,40)(32,33)(35,38)(36,39);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28)(26,31)(27,35)(29,38)(30,39)(32,36)(33,40)(34,37);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^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, 8761423);