Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
30712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 50 points:
(1,2);(4,8)(6,14)(9,11)(12,18)(20,25)(26,36)(27,37)(28,38)(39,47)(40,48)(41,42)(49,50);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16)(20,25)(21,29)(27,37)(28,38)(30,43)(31,44)(41,42)(45,46);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17)(19,20)(21,26)(22,27)(23,28)(24,25)(29,36)(30,39)(31,40)(32,41)(33,37)(34,42)(35,38)(43,47)(44,48)(45,49)(46,50);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)(19,21)(20,26)(22,30)(23,31)(24,29)(25,36)(27,39)(28,40)(32,45)(33,43)(34,46)(35,44)(37,47)(38,48)(41,49)(42,50);(19,22,24,33)(20,27,25,37)(21,30,29,43)(23,34,35,32)(26,39,36,47)(28,42,38,41)(31,46,44,45)(40,50,48,49);(19,23,24,35)(20,28,25,38)(21,31,29,44)(22,32,33,34)(26,40,36,48)(27,41,37,42)(30,45,43,46)(39,49,47,50)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493590);