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 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,13,12,16)(14,17,22,24)(19,21,20,23)(25,26)(27,28)(29,37)(30,31)(32,40)(33,36)(34,39)(35,38);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,18,13)(14,19,22,20)(17,23,24,21)(25,27,30,33)(26,28,31,36)(29,38,40,34)(32,39,37,35);(2,5)(6,7)(14,22)(17,24)(19,20)(21,23)(26,31)(27,33)(32,37)(34,38);(3,8)(6,7)(9,12,15,11)(10,13,18,16)(14,20,22,19)(17,21,24,23)(25,28)(26,27)(29,39)(30,36)(31,33)(32,38)(34,37)(35,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,18)(14,21)(15,16)(17,20)(19,24)(22,23)(25,29,30,40)(26,32,31,37)(27,34,33,38)(28,35,36,39);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8254617);