Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C32 ⋊ #459 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#626 |
| commutator, cosocle | C2×C32×C4 | C23 |
| Frattini | C2×C4 | C2×D32 |
| Fitting | C32×C42 | C22 |
| socle | C22×C32 | C2×D4 |
| hypercenter | C42 | D32 |
| hypocenter | C32 | #459 |
| LCS | G ▷ C2×C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
327 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 74 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|b^2=df,c^2=e,d^2=f,b^a=bd,c^a=ce,d^a=df,h^a=h^2,c^b=ce,g^b=g^269 conjugacy classes — view the full character table.
| maximal subgroup of | #11778898, #11778901, #11778904, … |
| maximal normal subgroup of | #11778898, #11778901, #11778904, … |
| maximal quotient of | #11780803, #11780892, #11780957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 2237);