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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,4,5)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,3,6,4)(2,5,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,7)(5,8)(10,15)(13,23)(16,21)(17,24)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(2,7)(3,4)(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,4,6,3)(2,8,7,5)(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38);(1,5,6,8)(2,3,7,4)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229521);