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×#54552 |
| 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);(3,4,8,6)(5,9,13,11)(7,14,15,10)(12,17,18,16)(19,20)(21,26)(22,27)(23,30)(24,25)(28,33)(29,32)(31,34)(36,40)(37,43)(38,46)(48,50);(4,6)(9,11)(10,14)(16,17)(20,25)(21,29)(22,31)(23,33)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,5,8,13)(4,9,6,11)(7,12,15,18)(10,16,14,17)(19,21,24,29)(20,26,25,32)(22,28,31,30)(23,27,33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,6,8,4)(5,11,13,9)(7,14,15,10)(12,17,18,16)(19,22,24,31)(20,27,25,34)(21,30,29,28)(23,32,33,26)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23)(20,28)(21,27)(22,32)(24,33)(25,30)(26,31)(29,34)(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, 8010922);