Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×D32 |
| commutator, socle | C2×C32 | C22×C8 |
| Fitting | C32×C16 | C22 |
| prime radical | #3798 | C4 |
| cosocle | C32×C8 | C23 |
| hypercenter | C16 | D32 |
| hypocenter | C32 | #504 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C8 ◁ C16 | |
182 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 68 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=e,e^2=f,b^a=bf,c^a=cf,h^a=h^2,c^b=cf,g^b=g^296 conjugacy classes — view the full character table.
| maximal subgroup of | #11772727, #11797581, #11797616, … |
| maximal normal subgroup of | #11772727, #11797581, #11797616, … |
| maximal quotient of | #11795900, #11795904, #11796008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 2111);