Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C32×D8 ⋊ C22 |
| Sylow | C32 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D4 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C32×D8 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | D8 | D32 |
| hypocenter | C32 | #575 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
555 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 555 conjugacy classes of subgroups — too many to draw. Only its 126 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:
2^6,3^2|e^2=f,d^a=df,h^a=h^2,c^b=cf,g^b=g^2,d^c=de,e^c=ef,e^d=ef51 conjugacy classes — view the full character table.
| maximal subgroup of | #11798288, #11798320, D3×#5773, … |
| maximal normal subgroup of | #11798288, #11798320, D3×#5773, … |
| maximal quotient of | #11857955, #11858510, #11860206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 6610);