Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | C32×2−1+4 ⋊ C2 |
| Sylow | C32 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#638 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C23×C3⋊D3 |
| Fitting | C32×2−1+4 | C2 |
| socle | C2×C32 | C22×D4 |
| hypercenter | 2−1+4 | C3⋊D3 |
| hypocenter | C32 | #578 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
744 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 219 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:
2^6,3^2|b^2=f,c^2=f,e^2=f,d^a=de,e^a=ef,g^a=g^2,h^a=h^2,c^b=cf,e^d=ef90 conjugacy classes — view the full character table.
| maximal subgroup of | #11886645, #11886648, #11886654, … |
| maximal normal subgroup of | #11886645, #11886648, #11886654, … |
| maximal quotient of | #11858053, #11858062, #11858573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8191);