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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,21)(22,24);(2,7)(3,4)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,26,30,31)(27,32,36,39)(28,33,37,29)(34,35,40,38);(1,3,6,4)(2,8,7,5)(13,21)(16,20)(18,24)(22,23)(26,31)(28,37)(32,39)(34,40);(1,4,6,3)(2,5,7,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,32)(28,34)(29,38)(30,36)(31,39)(33,35)(37,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28)(26,33)(27,34)(29,31)(30,37)(32,35)(36,40)(38,39);(9,11)(10,17)(12,15)(13,18)(14,19)(16,23)(20,22)(21,24)(25,29,30,33)(26,28,31,37)(27,35,36,38)(32,40,39,34)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8456749);