Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9336 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,8,12,9,14,7,13)(5,15,18,25,19,28,17,10)(6,20,22,11,23,26,21,27)(16,24,31,33,32,34,30,29)(35,36)(37,46,49,57)(38,50)(39,44)(40,54)(41,45)(42,48,58,53)(43,51)(47,62,65,56)(52,59)(55,60)(61,63,66,64);(3,5,9,19)(4,10)(6,16,23,32)(7,17,8,18)(11,24)(12,15)(13,28)(14,25)(20,29)(21,30,22,31)(26,33)(27,34)(35,37,39,53,41,49,40,48)(36,42,44,57,45,58,54,46)(38,47,51,64,52,65,55,63)(43,56,59,66,60,62,50,61);(3,6,9,23)(4,11,14,27)(5,16,19,32)(7,22,8,21)(10,24,25,34)(12,20,13,26)(15,29,28,33)(17,31,18,30)(35,38)(36,43)(37,47)(39,51)(40,55)(41,52)(42,56)(44,59)(45,60)(46,61)(48,63)(49,65)(50,54)(53,64)(57,66)(58,62)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106763);