Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,21,17,19,24,18)(15,20,22,27,23,26,28,25);(5,7,9,8)(6,10,11,12)(14,18)(15,22)(16,24)(19,21)(23,28)(25,27);(5,8,9,7)(6,10,11,12)(14,19)(18,21)(20,26)(25,27);(13,15,16,22,17,23,24,28)(14,20,21,27,19,26,18,25);(14,19)(15,23)(18,21)(22,28)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287873);