Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | D4×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,13,16)(11,17,19,23)(12,21,22,15)(14,18,24,20);(9,11)(10,14)(12,18)(13,19)(15,23)(16,24)(17,21)(20,22);(2,5)(3,8)(6,7);(11,18)(12,22)(14,23)(15,21)(17,24)(19,20);(11,19)(14,24)(17,23)(18,20)Polycyclic presentation:
2^9|a^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237138, #72321187 |
| maximal normal subgroup of | #72237138, #72321187 |
| maximal quotient of | #102×AGL(1,ℤ/8), #72279888, #72309444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265925);