Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C9 ⋊ C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D9 |
| commutator | C2×C9 | C2×C42 |
| Frattini, socle, prime radical | C23×C3 | C22×D3 |
| Fitting | C9×#95 | C2 |
| cosocle | C23×C9 | C23 |
| hypercenter | #95 | D9 |
| hypocenter | C9 | C4×C4⋊C4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ #95 | |
291 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^2|a^2=e,b^2=f,c^2=d,g^3=h^2,g^a=g^2h,h^a=h^2,c^b=cef120 conjugacy classes — view the full character table.
| maximal subgroup of | #11775042, #11775071, #11775248, … |
| maximal normal subgroup of | #11775042, #11775071, #11775248, … |
| maximal quotient of | #95×Dic9, #11773171, Dic9⋊#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 211);