Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C72 ⋊ SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11716032 |
| commutator, Fitting, cosocle | C72×C8 | C22 |
| Frattini, hypercenter | C8 | D72 |
| socle | C2×C72 | D8 |
| prime radical | C7×D56 | C2 |
| hypocenter | C72 | SD32 |
| LCS | G ▷ C72×C8 ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
67 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^5,7^2|b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,g^a=g^6,c^b=cd,d^b=de,f^b=f^6125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 205);