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 | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,24)(20,21)(26,29)(27,33)(32,37)(34,39);(1,2)(3,7)(4,8)(5,6)(10,16)(11,13)(18,22)(19,21)(28,36)(32,37)(34,39)(38,40);(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,3,5,4)(2,7,6,8)(9,12,15,17)(10,11,16,13)(14,20,23,24)(18,19,22,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,29,30,26)(27,35,33,31)(28,37,36,32)(34,40,39,38)Polycyclic presentation:
2^9|c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549155);