Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C32×#41 ⋊ C22 |
| Sylow | C32 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3811 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C32×#41 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | #41 | D32 |
| hypocenter | C32 | C2×#135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
614 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|c^2=f,e^2=f,d^a=de,e^a=ef,h^a=h^2,d^b=de,e^b=ef,g^b=g^2,e^d=ef66 conjugacy classes — view the full character table.
| maximal subgroup of | #11787411, #11787413, #11861003, … |
| maximal normal subgroup of | #11787411, #11787413, #11861003, … |
| maximal quotient of | #11857788, #11857790, #11858430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7341);