Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533135 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62901 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,21,23,16)(12,22,24,17)(13,18)(14,19)(15,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,17)(14,22)(15,23)(18,24)(20,21);(2,5)(6,8)(9,12)(10,17)(11,19)(13,16)(14,21)(15,24)(18,23)(20,22);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,18,20,19)(11,17,23,22)(12,16,24,21);(3,7)(6,8)(9,14,15,13)(10,19,20,18)(11,22,23,17)(12,21,24,16)Polycyclic presentation:
2^9|a^2=fg,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449435);