Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55968 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,9)(6,14)(7,8)(10,15)(11,12)(13,18)(16,17)(19,20,23,26)(21,27,28,24)(22,25,30,32)(29,33,34,31)(36,40)(37,43)(38,46)(48,50);(4,8)(5,11)(6,15)(16,18)(21,28)(24,27)(29,34)(31,33)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21)(20,24)(22,29)(23,28)(25,31)(26,27)(30,34)(32,33)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,5,7,11)(4,9,8,12)(6,13,15,17)(10,16,14,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,6)(4,10)(5,13)(7,15)(8,14)(9,16)(11,17)(12,18)(19,22,23,30)(20,25,26,32)(21,29,28,34)(24,31,27,33)(35,38,39,46)(36,42,40,47)(37,45,43,49)(41,48,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010139);