Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C23×C32 ⋊ D4 |
| Sylow | C32 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×GO+(4,𝔽2) |
| commutator, cosocle | C22×C3⋊D3 | C23 |
| Fitting | C23×C32 | D4 |
| socle | C22×C32 | C2×D4 |
| hypercenter | C23 | GO+(4,𝔽2) |
| hypocenter | C32 | #392 |
| LCS | G ▷ C22×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C23 | |
558 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|b^a=bd,c^a=cf,g^a=g^2,c^b=ce,g^b=h,h^b=g,g^d=g^2,h^d=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | #11771795, #11778040, C2×#10593530, … |
| maximal normal subgroup of | #11771795, #11778040, C2×#10593530, … |
| maximal quotient of | #11773287, #11773289, #11773320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5412);