Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C72 · Schur multiplier (H2) = C22×C7 |
| direct product | C2×C72×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C72 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C72 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C72 ◁ C2×C4×C72 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 1568 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(19,20,21,22,23,24,25);(3,4,5,6,7,8,9,10)(11,18,17,16,15,14,13,12)(26,27,28,29,30,31,32);(3,11,7,15)(4,12,8,16)(5,13,9,17)(6,14,10,18)686 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 780);