Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61289 |
| 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 | |
6474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12)(13,18)(14,20)(15,22)(16,21)(17,23)(19,27)(24,32)(25,37)(26,34)(28,36)(29,33)(30,35)(31,39)(38,42)(40,41);(3,5)(4,8)(6,9)(7,10)(11,13,17,26)(12,18,23,34)(14,24,28,40)(15,29,16,25)(19,31,35,42)(20,32,36,41)(21,37,22,33)(27,39,30,38);(3,6,7,4)(5,9,10,8)(11,14)(12,19)(13,24)(15,27)(16,30)(17,28)(18,31)(20,21)(22,36)(23,35)(25,38)(26,40)(29,39)(32,37)(33,41)(34,42);(5,10)(8,9)(11,14)(12,20)(13,24)(15,30)(16,27)(17,28)(18,32)(19,21)(22,35)(23,36)(25,39)(26,40)(29,38)(31,37)(33,42)(34,41);(11,15,17,16)(12,21,23,22)(13,25,26,29)(14,27,28,30)(18,33,34,37)(19,20,35,36)(24,38,40,39)(31,41,42,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483802);