Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C257 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #16861632 |
| commutator, hypocenter | C257 | C32 |
| Fitting, socle | C2×C257 | C16 |
| LCS | G ▷ C257 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^249,f^b=f^64,f^c=f^241,f^d=f^25664 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22551691, #30657866, #30657867, … |
| maximal normal subgroup of | C2×#22551691, #30657866, #30657867, … |
| maximal quotient of | #30657861, C2×#22551691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 245);