Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #492504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | D4×D16 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(10,14)(11,17)(12,19)(15,22)(16,23)(18,21)(20,24);(1,2,5,7)(3,6,4,8)(10,15)(14,22)(16,21)(18,23);(9,10)(11,18)(12,21)(13,15)(14,17)(16,19)(20,23)(22,24);(1,3)(4,5)(6,8)Polycyclic presentation:
2^9|b^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg59 conjugacy classes — view the full character table.
| maximal subgroup of | #72226906, #72309684, SD16×#509, … |
| maximal normal subgroup of | #72226906, #72309684, SD16×#509, … |
| maximal quotient of | #72226890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399782);