Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×M32 ⋊ C2 |
| Sylow | C9 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D72 |
| commutator, cosocle | C8×C9 | C23 |
| Frattini | C3×C8 | C22×D3 |
| Fitting | C9×M32 | C2 |
| socle | C2×C3 | C2×D24 |
| hypercenter | M32 | D9 |
| hypocenter | C9 | #509 |
| LCS | G ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32 | |
135 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 47 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|c^2=de,d^2=ef,e^2=f,g^3=h^2,c^a=cd,d^a=de,e^a=ef,g^a=g^2h,h^a=h^2,c^b=cf96 conjugacy classes — view the full character table.
| maximal subgroup of | #11797949, #11797958, #11798269, … |
| maximal normal subgroup of | #11797949, #11797958, #11798269, … |
| maximal quotient of | #11796678, #11796708, #11797219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 470);