Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C3×#33996 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C2 | C23×C3×C42 |
| Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
3630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,6,7)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27);(8,9,10,11)(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(14,21)(17,20)(22,27)(24,26);(13,18)(17,20)(19,25)(24,26);(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26984);