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 |
| prime radical | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,20)(11,14,19,16)(12,18,13,17)(21,24,22,23)(25,26)(27,34)(28,29)(30,33)(31,35)(32,38)(36,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,27)(26,32)(28,39)(29,37)(30,36)(31,38)(33,40)(34,35);(9,12,15,13)(10,17,20,18)(11,21,19,22)(14,23,16,24)(25,28,31,30)(26,29,35,33)(27,36,38,39)(32,40,34,37);(1,3,5,4)(2,6,8,7)(9,13,15,12)(10,18,20,17)(11,22,19,21)(14,24,16,23)(25,29,31,33)(26,28,35,30)(27,37,38,40)(32,39,34,36);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,21)(13,22)(15,19)(16,20)(17,23)(18,24)(25,30,31,28)(26,33,35,29)(27,36,38,39)(32,40,34,37);(9,14)(10,19)(11,20)(12,23)(13,24)(15,16)(17,22)(18,21)(25,27)(26,34)(28,36)(29,37)(30,39)(31,38)(32,35)(33,40)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562167);