Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #518 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator | C2×C32 | C23×C4 |
| Fitting, prime radical | C22×C32×D4 | C2 |
| socle, cosocle | C22×C32 | C24 |
| hypercenter | C22×D4 | C3⋊D3 |
| hypocenter | C32 | #518 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
1014 conjugacy classes of subgroups; only the 471 normal subgroups are shown.
This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 471 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^2|a^2=e,d^a=df,g^a=g^2,h^a=h^2,c^b=cf114 conjugacy classes — view the full character table.
| maximal subgroup of | #11785896, #11785910, #11785915, … |
| maximal normal subgroup of | #11785896, #11785910, #11785915, … |
| maximal quotient of | #11800856, #11800876, #11800906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8151);