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 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(25,26,32,29)(27,36,38,33)(28,39,31,34)(30,40,37,35);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,27,32,38)(26,33,29,36)(28,37,31,30)(34,35,39,40);(10,15)(12,16)(18,21)(20,23)(25,28,32,31)(26,34,29,39)(27,37,38,30)(33,35,36,40);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,29,32,26)(27,36,38,33)(28,34,31,39)(30,40,37,35);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,30)(26,35)(27,28)(29,40)(31,38)(32,37)(33,39)(34,36);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,31,32,28)(26,34,29,39)(27,30,38,37)(33,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7635427);