Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C9×H(5,𝔽2) ⋊ C2 |
| Sylow | C9 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#611 |
| commutator, cosocle | C4×C9 | C24 |
| Frattini | C3×C4 | C23×D3 |
| Fitting | C9×H(5,𝔽2) | C2 |
| socle | C2×C3 | C22×#67 |
| hypercenter | H(5,𝔽2) | D9 |
| hypocenter | C9 | #577 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
387 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 135 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|b^2=f,c^2=f,d^2=f,e^2=f,g^3=h,d^a=de,e^a=ef,g^a=g^2h^2,h^a=h^2,c^b=cf,e^d=ef90 conjugacy classes — view the full character table.
| maximal subgroup of | #11886556, #11886565, #11886653, … |
| maximal normal subgroup of | #11886556, #11886565, #11886653, … |
| maximal quotient of | #11858005, #11858052, #11858533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1820);