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 | |
4321 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)(11,19)(12,23)(13,24)(14,18)(15,16)(17,20)(21,22)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,4)(5,6)(7,8)(10,16)(11,20)(12,24)(18,22)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(2,6)(4,7)(9,11,15,20)(10,17,16,19)(12,21,24,14)(13,22,23,18)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,12)(10,13)(11,21)(14,20)(15,24)(16,23)(17,22)(18,19)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,12)(11,22)(14,19)(15,23)(16,24)(17,21)(18,20);(1,4,5,7)(2,3,6,8)(9,14,15,21)(10,18,16,22)(11,12,20,24)(13,19,23,17)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8138632);