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 | |
4518 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)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(26,31)(27,35)(28,38)(34,39);(1,2)(3,4)(5,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,26)(27,32)(28,33)(29,39)(30,31)(34,36)(35,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,27,30,35)(26,32,31,37)(28,29,38,36)(33,39,40,34);(3,8)(4,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28,30,38)(26,33,31,40)(27,36,35,29)(32,34,37,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29,30,36)(26,34,31,39)(27,28,35,38)(32,40,37,33);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,26)(27,37)(28,33)(29,34)(30,31)(32,35)(36,39)(38,40)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8875756);