Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
33682 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);(4,7)(8,10)(12,13)(16,29)(18,21)(19,22)(25,38)(27,39)(30,33)(36,42);(3,4)(5,8)(6,7)(9,10)(11,12,17,13)(14,18,26,21)(15,22,28,19)(16,20,29,23)(24,33,37,30)(25,31,38,34)(27,35,39,32)(36,41,42,40);(3,5,6,9)(4,8,7,10)(11,13,17,12)(14,18,26,21)(15,22,28,19)(16,23,29,20)(24,30,37,33)(25,31,38,34)(27,35,39,32)(36,40,42,41);(11,14)(12,18)(13,21)(15,24)(16,25)(17,26)(19,30)(20,31)(22,33)(23,34)(27,36)(28,37)(29,38)(32,40)(35,41)(39,42);(11,15,17,28)(12,19,13,22)(14,24,26,37)(16,27,29,39)(18,30,21,33)(20,32,23,35)(25,36,38,42)(31,40,34,41);(11,16)(12,20)(13,23)(14,25)(15,27)(17,29)(18,31)(19,32)(21,34)(22,35)(24,36)(26,38)(28,39)(30,40)(33,41)(37,42)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493213);