Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| 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 | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,19,23,20,22,24,21);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(18,21)(19,24)(22,23);(2,6)(5,8)(10,14)(11,15)(18,22)(21,23);(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6512770);