Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62886 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
33854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(12,17)(13,21)(15,28)(19,25)(23,33)(26,39)(31,36)(40,42);(4,7)(8,9)(11,12,16,17)(13,18,21,22)(14,25,27,19)(15,20,28,29)(23,35,33,30)(24,31,34,36)(26,38,39,32)(37,42,41,40);(3,4)(5,8)(6,7)(9,10)(11,12)(13,22)(14,19)(15,20)(16,17)(18,21)(23,35)(24,36)(25,27)(26,32)(28,29)(30,33)(31,34)(37,42)(38,39)(40,41);(3,5)(4,8)(6,10)(7,9)(11,13)(12,18)(14,23)(15,24)(16,21)(17,22)(19,30)(20,31)(25,35)(26,37)(27,33)(28,34)(29,36)(32,40)(38,42)(39,41);(11,14,16,27)(12,19,17,25)(13,23,21,33)(15,26,28,39)(18,30,22,35)(20,32,29,38)(24,37,34,41)(31,40,36,42);(11,15)(12,20)(13,24)(14,26)(16,28)(17,29)(18,31)(19,32)(21,34)(22,36)(23,37)(25,38)(27,39)(30,40)(33,41)(35,42)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493417);