Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C24 |
| Fitting | C32×D4 | D4 |
| socle | C2×C32 | C22×D4 |
| hypercenter | D4 | GO+(4,𝔽2) |
| hypocenter | C32 | #576 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
434 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|e^2=f,b^a=be,e^a=ef,g^a=g^2,e^b=ef,g^b=h,h^b=g,d^c=df,g^e=g^2,h^e=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | #11886640, #11886641, #11886830, … |
| maximal normal subgroup of | #11886640, #11886641, #11886830, … |
| maximal quotient of | #11858043, #11858602, #11860553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8396);