Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C241 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#11757210 |
| commutator, socle | C2×C241 | C2×C8 |
| Fitting | C8×C241 | C4 |
| hypercenter | C8 | #11757210 |
| hypocenter | C241 | M32 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C4 ◁ C8 | |
26 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=c,c^2=d,d^2=e,b^a=be,f^a=f^177,f^b=f^240,f^c=f^240500 conjugacy classes — view the full character table.
| maximal subgroup of | #29508607, #29508608, #29508633, … |
| maximal normal subgroup of | #29508607, #29508608, #29508633, … |
| maximal quotient of | #29508606, #29508612, #29508638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 221);