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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7911 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|c^2=g,f^2=i,h^2=i,b^a=bg,c^a=ch,h^a=hi,d^c=dh,e^c=ei,h^c=hi,h^d=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72237198, #72249499, #72249629, … |
| maximal normal subgroup of | #72237198, #72249499, #72249629, … |
| maximal quotient of | #72237210, #72256968, #72256977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483537);