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 | #46133 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,21,24,18)(25,26)(27,31)(28,34)(29,33)(30,32)(35,37)(36,40)(38,39);(1,2,4,6)(3,5,8,7)(10,15)(13,24)(16,19)(17,22)(26,27)(29,37)(32,34)(36,38);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(17,22)(18,21)(25,27,31,26)(28,34,30,32)(29,35,37,33)(36,40,38,39);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,17,24,22)(25,28,31,30)(26,32,27,34)(29,36,37,38)(33,39,35,40);(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29,31,37)(26,33,27,35)(28,36,30,38)(32,39,34,40);(9,13,14,24)(10,18,15,21)(11,17,20,22)(12,19,23,16)(25,30,31,28)(26,34,27,32)(29,38,37,36)(33,40,35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664611);