Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D24 |
| commutator, cosocle | C32×C8 | C23 |
| Frattini | C8 | C2×D32 |
| Fitting | C32×C16 | C22 |
| socle | C2×C32 | C2×D8 |
| hypercenter | C16 | D32 |
| hypocenter | C32 | #508 |
| LCS | G ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
182 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 48 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^2=f,c^2=de,d^2=ef,e^2=f,b^a=bf,h^a=h^2,c^b=cd,d^b=de,e^b=ef,g^b=g^281 conjugacy classes — view the full character table.
| maximal subgroup of | #11800565, #11800570, #11800573, … |
| maximal normal subgroup of | #11800565, #11800570, #11800573, … |
| maximal quotient of | #11796657, #11797201, #11797643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 2125);