Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63912 |
| commutator, cosocle | C2×C32×C4 | C23 |
| Frattini | C2×C4 | C2×D32 |
| Fitting | C32×M16 | C22 |
| socle | C2×C32 | #121 |
| hypercenter | M16 | D32 |
| hypocenter | C32 | #472 |
| LCS | G ▷ C2×C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
247 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 52 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|c^2=d,d^2=f,b^a=bd,c^a=ce,d^a=df,h^a=h^2,c^b=ce,d^b=df,g^b=g^2,e^c=ef48 conjugacy classes — view the full character table.
| maximal subgroup of | #11798369, #11798416, #11799403, … |
| maximal normal subgroup of | #11798369, #11798416, #11799403, … |
| maximal quotient of | #11780900, #11780932, #11781190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 2870);