Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C9×D8 ⋊ C2 |
| Sylow | C9 ⋊ #496 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D9 |
| commutator, cosocle | C2×C4×C9 | C23 |
| Frattini | C2×C3×C4 | C22×D3 |
| Fitting | C2×C9×D8 | C2 |
| socle | C22×C3 | D3×D4 |
| hypercenter | C2×D8 | D9 |
| hypocenter | C9 | #496 |
| LCS | G ▷ C2×C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^2|a^2=e,d^2=f,g^3=h^2,b^a=be,c^a=cef,g^a=g^2h,h^a=h^2,c^b=cd,d^b=df,d^c=df72 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#496, C2×#10588834, #11874413, … |
| maximal normal subgroup of | D9×#496, C2×#10588834, #11874413, … |
| maximal quotient of | #11782617, #11782623, #11782657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 716);