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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,15,19)(11,16,20,14)(12,17,21,24)(13,18,22,23)(25,26,31,34)(27,35,32,28)(29,30,33,40)(36,37,39,38);(1,2,5,8)(3,6,4,7)(9,11,15,20)(10,16,19,14)(12,22,21,13)(17,23,24,18)(25,27,31,32)(26,28,34,35)(29,39,33,36)(30,37,40,38);(1,3,5,4)(2,6,8,7)(11,20)(13,22)(14,16)(18,23)(29,33)(30,40)(36,39)(37,38);(9,12)(10,17)(11,13)(14,23)(15,21)(16,18)(19,24)(20,22)(25,28,31,35)(26,32,34,27)(29,37,33,38)(30,36,40,39);(1,2)(3,6)(4,7)(5,8)(9,13)(10,18)(11,21)(12,20)(14,17)(15,22)(16,24)(19,23)(25,29,31,33)(26,30,34,40)(27,36,32,39)(28,38,35,37);(1,4,5,3)(2,7,8,6)(9,14)(10,11)(12,23)(13,17)(15,16)(18,21)(19,20)(22,24)(25,30)(26,33)(27,37)(28,36)(29,34)(31,40)(32,38)(35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948203);