Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23 |
| Frattini, socle | C2×C3 | C22×S4 |
| Fitting | C3×H(5,𝔽2) | D3 |
| hypercenter | Q8 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
256 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^3,3^2,2^3|b^2=h,c^2=h,d^3=e,f^2=h,g^2=h,c^a=ch,d^a=d^2e^2,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^d=g,g^d=fgh,g^f=gh42 conjugacy classes — view the full character table.
| maximal subgroup of | #11920366, #11920367, #11920369, … |
| maximal normal subgroup of | #11920366, #11920367, #11920369, … |
| maximal quotient of | #11920263, #11920265, #11920266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4982);