Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of #64188 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊(C22⋊C9) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #64188 |
| commutator, hypocenter | C22⋊(C22⋊C9) | C4 |
| Frattini | C2×C3 | #1020 |
| Fitting, socle, prime radical | C25×C3 | D3 |
| cosocle | C2×C22⋊(C22⋊C9) | C2 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 ◁ C2 | |
283 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^2,2^4|a^2=b,c^3=d,c^a=c^2d^2,d^a=d^2,e^a=f,f^a=e,g^a=h,h^a=g,e^c=f,f^c=ef,g^c=gh,h^c=g36 conjugacy classes — view the full character table.
| maximal subgroup of | #11919944, #64188⋊C4, #11920034, … |
| maximal normal subgroup of | #11919944, #64188⋊C4, #11920034, … |
| maximal quotient of | #11920460, #11920461, #11920462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4993);