Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6386 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);(4,7)(8,9)(11,12,16,21)(13,23,25,18)(14,27,29,36)(15,22,17,20)(19,31,34,28)(24,38,39,42)(26,33,30,35)(32,41,37,40);(3,4,6,7)(5,9,10,8)(11,13,16,25)(12,18,21,23)(14,24,29,39)(15,30,17,26)(19,32,34,37)(20,35,22,33)(27,42,36,38)(28,41,31,40);(11,14)(12,19)(13,24)(15,28)(16,29)(17,31)(18,32)(20,27)(21,34)(22,36)(23,37)(25,39)(26,40)(30,41)(33,38)(35,42);(3,5,6,10)(4,8,7,9)(13,25)(18,23)(24,39)(26,30)(32,37)(33,35)(38,42)(40,41);(11,15,16,17)(12,20,21,22)(13,26,25,30)(14,28,29,31)(18,33,23,35)(19,27,34,36)(24,40,39,41)(32,38,37,42)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483706);