Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(10,16)(11,19,21,17)(12,14)(13,18)(20,24,23,22);(5,6,7,8)(9,10,12,16,15,18,14,13)(11,17,20,24,21,19,23,22);(9,11,12,20,15,21,14,23)(10,17,16,24,18,19,13,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,g^a=gi,h^a=hi,e^c=eh,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12823, #72242481 |
| maximal normal subgroup of | C4×#12823, #72242481 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54084);