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 | #2656 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,31,35)(27,34,29,32)(28,30,39,33)(36,38,37,40);(1,2,4,5)(3,6,7,8)(25,27,31,29)(26,32,35,34)(28,37,39,36)(30,38,33,40);(10,15)(11,17)(16,22)(18,23)(25,28,31,39)(26,33,35,30)(27,36,29,37)(32,38,34,40);(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28)(26,30)(27,37)(29,36)(31,39)(32,38)(33,35)(34,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,27)(26,34,35,32)(28,36,39,37)(30,40,33,38);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30)(26,28)(27,38)(29,40)(31,33)(32,37)(34,36)(35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618653);