Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1455 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,28,31)(27,33,36,30)(29,32,35,38)(34,39,40,37);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40);(1,3,5,4)(2,6,7,8)(25,27,29,34,28,36,35,40)(26,30,32,37,31,33,38,39);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(22,23)(25,27,35,40,28,36,29,34)(26,30,38,39,31,33,32,37)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=ghi,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744355);