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 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10,15,18)(11,13,17,23)(12,14,22,16)(19,24,20,21)(25,26)(27,34)(28,32)(29,39)(30,35)(31,36)(33,37)(38,40);(1,2,6,5)(3,7,4,8)(9,11,15,17)(10,13,18,23)(12,20,22,19)(14,21,16,24)(25,27)(26,30)(28,37)(29,38)(31,35)(32,39)(33,40)(34,36);(1,2,6,5)(3,8,4,7)(9,12,15,22)(10,16,18,14)(11,19,17,20)(13,21,23,24)(25,28,31,38)(26,32,36,40)(27,37,35,29)(30,39,34,33);(1,3,6,4)(2,7,5,8)(9,12)(10,14)(11,20)(13,21)(15,22)(16,18)(17,19)(23,24)(25,29,31,37)(26,33,36,39)(27,38,35,28)(30,40,34,32);(1,4,6,3)(2,7,5,8)(9,13,15,23)(10,17,18,11)(12,21,22,24)(14,19,16,20)(25,27)(26,34)(28,37)(29,38)(30,36)(31,35)(32,33)(39,40);(1,5,6,2)(3,8,4,7)(9,14)(10,12)(11,21)(13,20)(15,16)(17,24)(18,22)(19,23)(25,30,31,34)(26,35,36,27)(28,39,38,33)(29,40,37,32)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592719);