Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C4×C257 ⋊ C4 |
| Sylow | C257 ⋊ #117 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#11759967 |
| commutator, socle | C2×C257 | C22×C4 |
| Frattini | C2 | C22×#11759967 |
| Fitting | C2×C4×C257 | C4 |
| hypercenter | C2×C4 | #11759967 |
| hypocenter | C257 | #117 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
76 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=e,c^2=d,b^a=bd,f^a=f^241,f^e=f^256532 conjugacy classes — view the full character table.
| maximal subgroup of | #30657604, #30657605, #30657608, … |
| maximal normal subgroup of | #30657604, #30657605, #30657608, … |
| maximal quotient of | #30657673, C4×C22⋊#11759967, C4×C4⋊#11759967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 225);