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 | |
4363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(26,31)(28,32)(33,39)(36,37);(1,2)(3,4)(5,6)(7,8)(12,19)(16,22)(18,23)(20,24)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(2,6)(3,7)(10,15)(11,17)(16,22)(18,23)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(3,7)(4,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,3)(2,4)(5,7)(6,8)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,e^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,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532676);