Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | C23×C32×C4 ⋊ C2 |
| Sylow | C32 ⋊ #525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Frattini | C22 | C23×C3⋊D3 |
| Fitting | C23×C32×C4 | C2 |
| hypercenter | C23×C4 | C3⋊D3 |
| hypocenter | C32 | #525 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 | |
990 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 990 conjugacy classes of subgroups — too many to draw. Only its 255 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|b^2=f,c^a=cf,d^a=de,g^a=g^2,h^a=h^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #11775010, #11775019, #11775031, … |
| maximal normal subgroup of | #11775010, #11775019, #11775031, … |
| maximal quotient of | #11801048, #11801478, #11801821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8144);