Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C257 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D257 |
| commutator | C2×C257 | C22×C4 |
| Fitting | C4⋊C4×C257 | C2 |
| socle | C22×C257 | C23 |
| hypercenter | C4⋊C4 | D257 |
| hypocenter | C257 | C4×Q8 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
70 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=e,b^2=d,c^2=de,b^a=bd,f^c=f^2561300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30656824, #30656828, #30656829, … |
| maximal normal subgroup of | #30656824, #30656828, #30656829, … |
| maximal quotient of | #30656588, C4⋊C4×Dic257, #30656618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 111);