Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C2×AGL(1,ℤ/8) | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,3,8,7)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24);(1,3)(2,5)(4,6)(7,8)(9,11,12,18,13,19,20,24,14,10,15,16,21,17,22,23);(1,2)(3,5)(4,6)(7,8)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,d^b=dg,f^b=fg,e^c=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#453, #72242854 |
| maximal normal subgroup of | C16×#453 |
| maximal quotient of | #72242854 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43951);