Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C64 · Schur multiplier (H2) = C1 |
| direct product | C3×M256 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C64 | C22 |
| commutator | C2 | C2×C3×C64 |
| Frattini, cosocle | C64 | C22×C3 |
| socle | C2×C3 | C2×C64 |
| prime radical | C22×C3 | C64 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C64 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 131 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(5,69)(7,71)(9,73)(11,75)(13,77)(15,79)(17,81)(19,83)(21,85)(23,87)(25,89)(27,91)(29,93)(31,95)(33,97)(35,99)(37,101)(39,103)(41,105)(43,107)(45,109)(47,111)(49,113)(51,115)(53,117)(55,119)(57,121)(59,123)(61,125)(63,127)(65,129)(67,131)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 538);