Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C9×#135 ⋊ C2 |
| Sylow | C9 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D9 |
| commutator, cosocle | C4×C9 | C24 |
| Frattini | C3×C4 | C23×D3 |
| Fitting | C9×#135 | C2 |
| socle | C2×C3 | C2×D3×D4 |
| hypercenter | #135 | D9 |
| hypocenter | C9 | #576 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
402 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 119 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|e^2=f,g^3=h,b^a=bf,g^a=g^2h^2,h^a=h^2,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef78 conjugacy classes — view the full character table.
| maximal subgroup of | #11798324, #11798336, #11798418, … |
| maximal normal subgroup of | #11798324, #11798336, #11798418, … |
| maximal quotient of | #11857993, #11858569, #11860460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1753);