Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1215 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26)(27,38)(28,34)(29,40)(30,36)(31,37)(32,35)(33,39);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,33)(28,29)(30,32)(31,39)(34,35)(36,40)(37,38);(2,6)(5,8)(25,28,29,27,31,30,32,39)(26,34,40,38,37,36,35,33);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,31,32)(26,35,37,40)(27,28,39,30)(33,34,38,36);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,32,27,31,28,29,39)(26,36,35,38,37,34,40,33)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721562);