Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61666 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,20)(14,27)(15,23)(16,22)(17,21)(18,26)(19,28)(24,38)(25,33)(29,37)(30,36)(31,35)(32,39)(34,41)(40,42);(3,5)(4,7)(6,10)(8,9)(11,13,16,26)(12,18,22,20)(14,24,29,40)(15,30,17,25)(19,32,35,34)(21,36,23,33)(27,42,37,38)(28,41,31,39);(11,14,16,29)(12,19,22,35)(13,24,26,40)(15,28,17,31)(18,32,20,34)(21,27,23,37)(25,39,30,41)(33,38,36,42);(5,10)(7,9)(11,13)(12,20)(14,24)(15,30)(16,26)(17,25)(18,22)(19,34)(21,33)(23,36)(27,38)(28,41)(29,40)(31,39)(32,35)(37,42);(4,8)(7,9)(11,15,16,17)(12,21,22,23)(13,25,26,30)(14,28,29,31)(18,33,20,36)(19,27,35,37)(24,39,40,41)(32,38,34,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484466);