Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #392×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2430 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator, socle | C23×C67 | C24 |
| Fitting | #392×C67 | C2 |
| hypercenter | #392 | D67 |
| hypocenter | C67 | #2430 |
| LCS | G ▷ C23×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ #392 | |
914 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg,h^d=h^66764 conjugacy classes — view the full character table.
| maximal subgroup of | #30829310, #30829314, #30829325, … |
| maximal normal subgroup of | #30829310, #30829314, #30829325, … |
| maximal quotient of | #30777794, #30778515, #30778692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 8113);