Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #518 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator | C2×C32 | C23×C4 |
| Fitting | C24×C32 | C22 |
| socle, cosocle | C22×C32 | C24 |
| prime radical | C22×#638 | C2 |
| hypercenter | C24 | D32 |
| hypocenter | C32 | #518 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C24 | |
794 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 200 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|a^2=e,b^2=e,d^a=df,h^a=h^2,c^b=cf,g^b=g^290 conjugacy classes — view the full character table.
| maximal subgroup of | #11785906, #11786596, #11787073, … |
| maximal normal subgroup of | #11785906, #11786596, #11787073, … |
| maximal quotient of | #11800862, #11800865, #11800893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7392);