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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(25,26,31,35)(27,36,38,32)(28,29,39,33)(30,40,37,34);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,31,38)(26,32,35,36)(28,37,39,30)(29,40,33,34);(1,3)(2,7)(4,5)(6,8)(10,14)(11,18)(12,21)(23,24)(25,28,31,39)(26,33,35,29)(27,37,38,30)(32,34,36,40);(3,4)(6,7)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,29,31,33)(26,28,35,39)(27,34,38,40)(30,36,37,32);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28)(26,29)(27,37)(30,38)(31,39)(32,40)(33,35)(34,36);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30,31,37)(26,34,35,40)(27,28,38,39)(29,36,33,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8966044);