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 | #2675 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 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)(9,10,15,17)(11,18,13,16)(12,14,21,24)(19,23,22,20)(29,37)(33,38)(34,39)(36,40);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,18)(12,19,21,22)(14,20,24,23)(25,26,30,31)(27,28,32,35)(29,33,37,38)(34,36,39,40);(10,17)(11,13)(12,21)(20,23)(26,31)(28,35)(33,38)(36,40);(9,12)(10,14)(11,19)(13,22)(15,21)(16,20)(17,24)(18,23)(25,27,30,32)(26,28,31,35)(29,34,37,39)(33,36,38,40);(9,13,15,11)(10,18,17,16)(12,22,21,19)(14,23,24,20)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(1,3,4,7)(2,6,5,8)(9,14)(10,12)(11,20)(13,23)(15,24)(16,19)(17,21)(18,22)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=hi,e^2=h,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, 7618065);