Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C67 ⋊ #1890 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14524012 |
| commutator | C23×C67 | C22×C4 |
| Frattini | C24 | C22×D67 |
| Fitting | #518×C67 | C2 |
| socle | C22×C67 | #120 |
| hypercenter | #518 | D67 |
| hypocenter | C67 | #1890 |
| LCS | G ▷ C23×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #518 | |
386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|a^2=f,b^2=dfg,b^a=bd,c^a=ce,e^a=eg,h^a=h^66,e^b=eg,d^c=dg,f^c=fg1154 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30771598, #30771600, #30771635, … |
| maximal normal subgroup of | #30771598, #30771600, #30771635, … |
| maximal quotient of | #30750521, #30750531, #30750810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4923);