Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C32×C4 ⋊ D4 |
| Sylow | C32 ⋊ #535 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C24 |
| Fitting | C2×C32×C4 | D4 |
| socle | C2×C32 | C22×D4 |
| hypercenter | C2×C4 | GO+(4,𝔽2) |
| hypocenter | C32 | #535 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
534 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^6,3^2|d^2=f,b^a=be,d^a=df,g^a=g^2,c^b=cf,g^b=h,h^b=g,g^e=g^2,h^e=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | #11788764, #11788797, #11788896, … |
| maximal normal subgroup of | #11788764, #11788797, #11788896, … |
| maximal quotient of | #11802665, #11802748, #11802999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8394);