Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C32 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C22×C32×C4 | C22 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #538 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
566 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 128 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|d^2=ef,b^a=be,c^a=cf,h^a=h^2,c^b=cf,d^b=df,g^b=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #11778198, #11778424, #11778650, … |
| maximal normal subgroup of | #11778198, #11778424, #11778650, … |
| maximal quotient of | #11802779, #11803056, #11803295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 6921);