Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907849 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,29,33)(27,34,30,31)(28,36,38,39)(32,35,40,37);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(25,27,29,30)(26,31,33,34)(28,37,38,35)(32,36,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,27,35,29,38,30,37)(26,32,31,39,33,40,34,36);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,27,29,30)(26,31,33,34)(28,35,38,37)(32,39,40,36);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819012);