Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C2×C4 |
| Frattini | C3×C4 | C2×S4 |
| Fitting | C3×#131 | D3 |
| socle | C2×C3 | C4×S4 |
| prime radical | #63691 | C2 |
| cosocle | #41⋊C9 | C22 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C8 | |
109 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 109 conjugacy classes of subgroups — too many to draw. Only its 22 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:
2^3,3^2,2^3|b^2=c,c^2=h,d^3=e^2,f^2=h,g^2=h,d^a=d^2e,e^a=e^2,f^a=gh,g^a=fh,f^d=g,g^d=fgh,g^f=gh60 conjugacy classes — view the full character table.
| maximal subgroup of | #11919994, #11919995, C2×#10589506, … |
| maximal normal subgroup of | #11919994, #11919995, C2×#10589506, … |
| maximal quotient of | C8×#3688, #11919964, C8×#3689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1388);