Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#38341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,8,7)(6,10,11,9)(22,26)(25,27);(5,7)(9,10)(13,16)(14,18)(20,21,24,23)(22,25,26,27);(12,13,15,16)(14,17,18,19)(20,22)(21,25)(23,27)(24,26);(4,6)(5,9)(7,10)(8,11)(20,21,24,23)(22,25,26,27);(4,7,8,5)(6,10,11,9)(12,14)(13,17)(15,18)(16,19)(20,23,24,21)(22,27,26,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 31329);