Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×D16 ⋊ C2 |
| Sylow | C9 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×D9 |
| commutator, cosocle | C8×C9 | C23 |
| Frattini | C3×C8 | C22×D3 |
| Fitting | C9×D16 | C2 |
| socle | C2×C3 | D3×D8 |
| hypercenter | D16 | D9 |
| hypocenter | C9 | #509 |
| LCS | G ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D16 | |
135 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^2|d^2=ef,e^2=f,g^3=h^2,c^a=cf,g^a=g^2h,h^a=h^2,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef60 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#509, C2×#10588591, #11899620, … |
| maximal normal subgroup of | D9×#509, C2×#10588591, #11899620, … |
| maximal quotient of | #11796666, #11796699, #11797210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 473);