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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,8)(6,7)(9,10)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(2,6)(3,8)(5,7);(2,7)(5,6)Polycyclic presentation:
2^9|b^2=i,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 | #135×AGL(1,ℤ/8), #72321187, #72342599 |
| maximal normal subgroup of | #135×AGL(1,ℤ/8), #72321187, #72342599 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265932);