Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62344 |
| 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 | |
6586 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)(19,31)(24,39)(25,34)(28,41)(29,37)(30,36)(32,38)(33,42)(35,40);(3,5)(4,7)(6,10)(8,9)(11,13,16,26)(12,18,22,20)(14,28,30,24)(15,32,17,25)(19,35,37,33)(21,38,23,34)(27,39,36,41)(29,40,31,42);(11,14,17,31,16,30,15,29)(12,19,23,27,22,37,21,36)(13,24,25,40,26,28,32,42)(18,33,34,41,20,35,38,39);(5,10)(7,9)(11,13)(12,20)(14,28)(15,32)(16,26)(17,25)(18,22)(19,33)(21,34)(23,38)(24,30)(27,41)(29,40)(31,42)(35,37)(36,39);(4,8)(7,9)(11,15,16,17)(12,21,22,23)(13,25,26,32)(14,29,30,31)(18,34,20,38)(19,36,37,27)(24,40,28,42)(33,41,35,39)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484833);