Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C22 | C24×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6717 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|d^2=i,e^2=g,h^2=i,b^a=bg,c^a=ch,h^a=hi,d^c=dh,f^c=fi,h^c=hi,h^d=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72242473, #72280658, #72282192, … |
| maximal normal subgroup of | #72242473, #72280658, #72282192, … |
| maximal quotient of | #72283561, #72283649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484160);