Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | C32×#135 ⋊ C2 |
| Sylow | C32 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×C3⋊D3 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C23×C3⋊D3 |
| Fitting | C32×#135 | C2 |
| socle | C2×C32 | C22×D4 |
| hypercenter | #135 | C3⋊D3 |
| hypocenter | C32 | #575 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
786 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 786 conjugacy classes of subgroups — too many to draw. Only its 179 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:
2^6,3^2|e^2=f,d^a=df,g^a=g^2,h^a=h^2,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef78 conjugacy classes — view the full character table.
| maximal subgroup of | #11798261, #11798270, #11798278, … |
| maximal normal subgroup of | #11798261, #11798270, #11798278, … |
| maximal quotient of | #11857936, #11858498, #11860184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8121);