Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #548 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×C22⋊C4 | C22 |
| hypercenter | C22⋊C4 | D32 |
| hypocenter | C32 | #548 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
627 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 627 conjugacy classes of subgroups — too many to draw. Only its 142 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|d^2=e,b^a=be,c^a=cf,h^a=h^2,c^b=cf,g^b=g^2,d^c=df75 conjugacy classes — view the full character table.
| maximal subgroup of | #11884042, #11884662, #11884816, … |
| maximal normal subgroup of | #11884042, #11884662, #11884816, … |
| maximal quotient of | C22⋊C4×#627, #11804930, #11805608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 6071);