Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#62161 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#529 ◁ G | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 51 points:
(1,2,3)(4,5,8,11)(6,12,13,9)(7,15,16,10)(14,17,19,18)(20,21)(22,27)(23,35)(24,29)(25,38)(26,30)(28,36)(31,45)(32,41)(33,42)(34,48)(37,43)(39,44)(40,46)(47,50)(49,51);(4,6)(5,9)(7,14)(8,13)(10,17)(11,12)(15,18)(16,19)(20,22)(21,27)(23,31)(24,32)(25,34)(26,33)(28,40)(29,41)(30,42)(35,45)(36,46)(37,47)(38,48)(39,49)(43,50)(44,51);(20,23,26,37)(21,28,30,44)(22,31,33,47)(24,36,25,39)(27,40,42,51)(29,43,38,35)(32,46,34,49)(41,50,48,45);(6,13)(9,12)(14,19)(17,18)(20,24,26,25)(21,29,30,38)(22,32,33,34)(23,36,37,39)(27,41,42,48)(28,43,44,35)(31,46,47,49)(40,50,51,45);(4,7,8,16)(5,10,11,15)(6,14,13,19)(9,17,12,18)(22,33)(27,42)(31,47)(32,34)(40,51)(41,48)(45,50)(46,49)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55149);