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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,20)(11,21,15,17)(12,23,22,18)(13,24,19,14)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2)(3,7)(4,5)(6,8)(9,11,16,15)(10,17,20,21)(12,19,22,13)(14,23,24,18)(26,27)(31,33)(32,34)(38,39);(2,5)(6,7)(9,12)(10,18)(11,13)(14,21)(15,19)(16,22)(17,24)(20,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,13,16,19)(10,14,20,24)(11,22,15,12)(17,18,21,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3,4,8)(2,6,5,7)(9,14)(10,19)(11,23)(12,21)(13,20)(15,18)(16,24)(17,22);(1,2)(3,6)(4,5)(7,8)(9,15,16,11)(10,17,20,21)(12,19,22,13)(14,18,24,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=g,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, 8121574);