Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle | C2×Q8⋊C9 | C22 |
| Frattini, socle | C22×C3 | C2×S4 |
| Fitting | C2×C3×#41 | D3 |
| hypercenter | C2×C4 | (C22⋊C9)⋊C2 |
| hypocenter | Q8⋊C9 | D4 |
| LCS | G ▷ C2×Q8⋊C9 ▷ Q8⋊C9 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
141 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3^2,2^3|b^2=h,d^3=e^2,f^2=h,g^2=h,b^a=bc,d^a=d^2e,e^a=e^2,f^a=gh,g^a=fh,f^d=g,g^d=fgh,g^f=gh51 conjugacy classes — view the full character table.
| maximal subgroup of | #11920242, #11920244, #11920247, … |
| maximal normal subgroup of | #11920242, #11920244, #11920247, … |
| maximal quotient of | #11920018, #11920020, #11920046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1411);