Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,30,34)(27,31,28,36)(29,38,39,33)(32,35,40,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,27)(26,31)(28,30)(29,35)(32,33)(34,36)(37,39)(38,40);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,27)(26,32,34,40)(29,35,39,37)(31,38,36,33);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(27,28)(31,36)(32,40)(35,37);(10,14)(15,21)(16,22)(23,24)(27,28)(31,36)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6860727);