Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C83 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #19127536 |
| commutator, cosocle | C32×C83 | C22 |
| Frattini | C32 | C2×D83 |
| Fitting | Q64×C83 | C2 |
| socle, prime radical | C2×C83 | D32 |
| hypercenter | Q64 | D83 |
| hypocenter | C83 | Q128 |
| LCS | G ▷ C32×C83 ▷ C16×C83 ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,83|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^82814 conjugacy classes — view the full character table.
| maximal subgroup of | Q128×D83, C2×#24905680, #35325456, … |
| maximal normal subgroup of | Q128×D83, C2×#24905680, #35325456, … |
| maximal quotient of | #35292284, #35292285, #35292303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 2727);