Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#12521 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C3 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C82 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,11)(6,12,15,19)(7,16,13,10)(9,17,18,14)(21,25)(22,26)(23,29)(27,35)(28,33)(30,34)(31,32);(4,6,13,17,8,15,7,14)(5,9,16,19,11,18,10,12);(20,21,26,32,23,27,30,33,24,28,34,35,29,31,22,25)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5509);