Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,21);(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,20,21,24);(5,6,7,8)(9,10,12,14)(11,13,16,15)(17,19,22,21)(18,24,23,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,21);(9,11,12,16)(10,13,14,15)(17,21,22,19)(18,24,23,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264315, #72264325, #72264485, … |
| maximal normal subgroup of | #72264315, #72264325, #72264485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836305);