Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#401 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ G | |
2942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(29,30,31);(3,4);(7,11)(9,12)(14,17)(18,19)(21,22,24,26)(23,27,28,25);(5,6,8,10)(7,12,11,9)(13,14,16,17)(15,19,20,18)(21,22)(23,27)(24,26)(25,28);(5,7)(6,9)(8,11)(10,12)(13,15,16,20)(14,18,17,19)(21,23,24,28)(22,25,26,27)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27536);