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×#62434 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6888 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);(4,7)(5,9)(11,12)(13,18)(14,26)(15,20)(16,30)(17,21)(19,28)(22,35)(23,32)(24,33)(25,39)(27,34)(29,40)(31,38)(36,41)(37,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,18)(14,22)(15,23)(16,25)(17,24)(19,31)(20,32)(21,33)(26,35)(27,36)(28,38)(29,37)(30,39)(34,41)(40,42);(11,14,15,27,17,29,16,28)(12,19,30,40,21,34,20,26)(13,22,23,36,24,37,25,38)(18,31,39,42,33,41,32,35);(4,7)(8,10)(11,15,17,16)(12,20,21,30)(13,23,24,25)(14,27,29,28)(18,32,33,39)(19,26,34,40)(22,36,37,38)(31,35,41,42);(3,5,6,9)(4,8,7,10)(13,24)(18,33)(22,37)(23,25)(31,41)(32,39)(35,42)(36,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486396);