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×#56010 |
| 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 | |
4066 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)(12,17)(19,20)(21,27)(22,24)(23,30)(25,29)(26,31)(28,32)(33,34)(35,36,41,39)(37,44,38,42)(40,47,48,43)(45,49,46,50);(4,8)(9,14)(10,15)(16,18)(20,22)(23,31)(25,27)(28,34)(35,37,41,38)(36,42,39,44)(40,45,48,46)(43,49,47,50);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,22,24,20)(21,25,29,27)(23,30,31,26)(28,32,34,33)(35,39,41,36)(37,42,38,44)(40,47,48,43)(45,49,46,50);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,40,41,48)(36,43,39,47)(37,45,38,46)(42,49,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010022);