Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C22×#41 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | C22⋊A4 |
| commutator, hypocenter | C22×Q8 | C2×C9 |
| Frattini | C2×C3 | C2×C22⋊A4 |
| Fitting, prime radical | C22×C3×#41 | C3 |
| socle | C23×C3 | C2×A4 |
| cosocle | C22×C3×Q8 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C3×C4 | |
334 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^2,2^5|a^2=h,b^3=c,f^2=h,g^2=h,d^b=e,e^b=de,f^b=g,g^b=fgh,g^f=gh72 conjugacy classes — view the full character table.
| maximal subgroup of | #1780⋊C9, #2388⋊C9, #2615⋊C9, … |
| maximal normal subgroup of | #1780⋊C9, #2388⋊C9, #2615⋊C9, … |
| maximal quotient of | C4×C22⋊(Q8⋊C9), #3459⋊C9, #41⋊(Q8⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5005);