Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61691 |
| 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 | |
7606 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)(11,12,16,22)(13,24,27,18)(14,20,29,36)(15,23,17,21)(19,31,35,28)(25,39,41,33)(26,34,30,37)(32,42,38,40);(3,5)(4,7)(6,10)(8,9)(11,13,16,27)(12,18,22,24)(14,25,29,41)(15,30,17,26)(19,32,35,38)(20,33,36,39)(21,37,23,34)(28,42,31,40);(11,14)(12,19)(13,25)(15,28)(16,29)(17,31)(18,32)(20,21)(22,35)(23,36)(24,38)(26,40)(27,41)(30,42)(33,37)(34,39);(5,10)(7,9)(11,14)(12,20)(13,25)(15,31)(16,29)(17,28)(18,33)(19,21)(22,36)(23,35)(24,39)(26,42)(27,41)(30,40)(32,37)(34,38);(4,8)(7,9)(11,15,16,17)(12,21,22,23)(13,26,27,30)(14,28,29,31)(18,34,24,37)(19,20,35,36)(25,40,41,42)(32,39,38,33)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484495);