Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C241 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #14515358 |
| commutator, hypocenter | C241 | C32 |
| Fitting | C4×C241 | C8 |
| socle | C2×C241 | C16 |
| LCS | G ▷ C241 | |
| UCS | C1 ◁ C4 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^30,f^b=f^177,f^c=f^240152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21440732, #29508824, #29508825, … |
| maximal normal subgroup of | C2×#21440732, #29508824, #29508825, … |
| maximal quotient of | #29508799, C2×#21440732 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 236);