Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4714 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)(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,24,23,20);(5,6)(7,8)(10,12)(14,15)(17,19,22,23)(18,20,21,24);(2,4)(5,6,7,8)(17,20,22,24)(18,23,21,19);(9,11)(10,14)(12,15)(13,16);(1,2)(3,4)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,20,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72264681, #72264694, #72264709, … |
| maximal normal subgroup of | #72264681, #72264694, #72264709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214170);