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 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,8)(6,7)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,2,4,7)(3,6,8,5);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,6)(3,8)(5,7);(2,7)(5,6)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^9|b^2=h,e^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 | #72306692, #72306700, #72306715, … |
| maximal normal subgroup of | #72306692, #72306700, #72306715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266000);