Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C8×C257 ⋊ C4 |
| Sylow | C257 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#11759967 |
| commutator, socle | C2×C257 | C42 |
| Fitting | C8×C257 | C4 |
| hypercenter | C8 | #11759967 |
| hypocenter | C257 | #97 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C4 ◁ C8 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=c,b^2=d,c^2=e,b^a=be,f^b=f^241,f^d=f^256532 conjugacy classes — view the full character table.
| maximal subgroup of | #30657627, #30657628, C2×#22551646, … |
| maximal normal subgroup of | #30657627, #30657628, C2×#22551646, … |
| maximal quotient of | #30657578, #30657579, #30657602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 200);