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×#62873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(20,25)(23,35)(26,29)(27,36)(31,45)(34,46)(38,41)(44,50);(3,4,7,10)(5,8,13,11)(6,9,14,16)(12,15,18,17)(22,33)(27,36)(30,43)(34,46)(38,41)(40,47)(44,50)(48,49);(3,4)(5,8)(6,9)(7,10)(11,13)(12,15)(14,16)(17,18)(19,20,24,25)(21,29,32,26)(22,27,33,36)(23,28,35,37)(30,41,43,38)(31,42,45,39)(34,40,46,47)(44,49,50,48);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21,24,32)(20,26,25,29)(22,30,33,43)(23,31,35,45)(27,38,36,41)(28,39,37,42)(34,44,46,50)(40,48,47,49);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22)(20,27)(21,30)(23,34)(24,33)(25,36)(26,38)(28,40)(29,41)(31,44)(32,43)(35,46)(37,47)(39,48)(42,49)(45,50);(19,23)(20,28)(21,31)(22,34)(24,35)(25,37)(26,39)(27,40)(29,42)(30,44)(32,45)(33,46)(36,47)(38,48)(41,49)(43,50)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493402);