Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C9 ⋊ #394 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D9 |
| commutator, cosocle | C23×C9 | C23 |
| Frattini, socle | C23×C3 | C22×D3 |
| Fitting | C22×D4×C9 | C2 |
| prime radical | C22×C3×D4 | D3 |
| hypercenter | C22×D4 | D9 |
| hypocenter | C9 | #394 |
| LCS | G ▷ C23×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
357 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^2|a^2=f,g^3=h^2,b^a=be,c^a=cd,g^a=g^2h,h^a=h^2,c^b=cf102 conjugacy classes — view the full character table.
| maximal subgroup of | #11778041, #11778047, #11778089, … |
| maximal normal subgroup of | #11778041, #11778047, #11778089, … |
| maximal quotient of | #11773277, #11773293, #11773325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 784);