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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,26,31,36)(27,34,29,32)(28,33,35,30)(37,38,39,40);(2,6)(3,7)(27,29)(32,34)(37,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,36,34)(28,37,35,39)(30,38,33,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,28)(26,33)(27,37)(29,39)(30,36)(31,35)(32,40)(34,38);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,29,31,27)(26,34,36,32)(28,39,35,37)(30,40,33,38);(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,30,31,33)(26,35,36,28)(27,38,29,40)(32,39,34,37)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10347171);