Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×H(5,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C3 |
| Fitting | C3×H(5,𝔽2) | C2×C3 |
| socle | C2×C3 | C23×A4 |
| hypercenter | Q8 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
401 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^2,3,2|b^2=h,c^2=h,e^2=h,f^2=h,c^a=ch,g^a=g^2,c^b=ch,e^d=ef,f^d=e,f^e=fh57 conjugacy classes — view the full character table.
| maximal subgroup of | #11922452, #11922454, #11922456, … |
| maximal normal subgroup of | #11922452, #11922454, #11922456, … |
| maximal quotient of | #85890⋊C3, #85885⋊C3, #85887⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8505);