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×#61264 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6458 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);(3,4,7,6)(5,9,10,8)(11,12)(13,18)(14,25)(15,20)(16,29)(17,21)(19,26)(22,37)(23,32)(24,34)(27,35)(28,33)(30,36)(31,39)(38,41)(40,42);(3,5)(4,8)(6,9)(7,10)(11,13,17,24)(12,18,21,34)(14,22,27,40)(15,28,16,23)(19,31,36,41)(20,33,29,32)(25,37,35,42)(26,39,30,38);(3,6,7,4)(5,9,10,8)(11,14)(12,19)(13,22)(15,26)(16,30)(17,27)(18,31)(20,35)(21,36)(23,38)(24,40)(25,29)(28,39)(32,37)(33,42)(34,41);(5,10)(8,9);(11,15,17,16)(12,20,21,29)(13,23,24,28)(14,26,27,30)(18,32,34,33)(19,35,36,25)(22,38,40,39)(31,37,41,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483785);