Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Sylow | C32 ⋊ H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C2×C32×D4 | C22 |
| hypercenter | C2×D4 | D32 |
| hypocenter | C32 | H(3,𝔽4) |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
558 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 120 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^a=bf,d^a=def,h^a=h^2,c^b=ce,g^b=g^2,d^c=df57 conjugacy classes — view the full character table.
| maximal subgroup of | #11780475, #11780584, #11780586, … |
| maximal normal subgroup of | #11780475, #11780584, #11780586, … |
| maximal quotient of | #11807599, #11807604, #11807892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7138);