Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#3802 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C32×M16 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | M16 | D32 |
| hypocenter | C32 | #578 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
527 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^2|b^2=f,c^2=f,d^2=e,e^2=f,d^a=de,e^a=ef,h^a=h^2,c^b=cf,d^b=de,e^b=ef,g^b=g^2,d^c=df54 conjugacy classes — view the full character table.
| maximal subgroup of | #11798416, #11798510, #11886690, … |
| maximal normal subgroup of | #11798416, #11798510, #11886690, … |
| maximal quotient of | #11858091, #11858595, #11860618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 6605);