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×#56753 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(4,8)(5,11)(10,15)(13,17)(19,20,24,27)(21,23,26,32)(22,25,31,34)(28,30,29,33)(36,40)(37,43)(38,46)(48,50);(6,14)(10,15)(13,17)(16,18)(21,26)(23,32)(28,29)(30,33)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,24,26)(20,23,27,32)(22,29,31,28)(25,33,34,30)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22)(20,25)(21,28)(23,30)(24,31)(26,29)(27,34)(32,33)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,6)(4,10)(5,13)(7,14)(8,15)(9,16)(11,17)(12,18)(19,23)(20,26)(21,27)(22,30)(24,32)(25,29)(28,34)(31,33)(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, 8007032);