Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C9 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D9 |
| commutator | C22×C9 | C22×C4 |
| Frattini, socle, prime radical | C23×C3 | C22×D3 |
| Fitting | C2×C42×C9 | C2 |
| cosocle | C23×C9 | C23 |
| hypercenter | C2×C42 | D9 |
| hypocenter | C9 | #389 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
255 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^2|a^2=f,b^2=e,c^2=d,g^3=h^2,b^a=be,c^a=cf,g^a=g^2h,h^a=h^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #11770664, #11770670, #11786820, … |
| maximal normal subgroup of | #11770664, #11770670, #11786820, … |
| maximal quotient of | #11773203, #11773225, #11773350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 490);