Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×D32 |
| commutator, socle, prime radical | C22×C32 | C22×C4 |
| Fitting | C2×C32×C8 | C22 |
| cosocle | C2×C32×C4 | C23 |
| hypercenter | C2×C8 | D32 |
| hypocenter | C32 | #446 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^2|a^2=e,b^2=e,c^2=d,d^2=f,b^a=be,c^a=cf,h^a=h^2,c^b=cf,g^b=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #11771708, #11777457, #11777555, … |
| maximal normal subgroup of | #11771708, #11777457, #11777555, … |
| maximal quotient of | #11784300, #11784342, #11788973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 2669);