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×#54659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4376 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);(4,8)(5,11)(10,15)(12,17)(19,20,23,26)(21,27,29,24)(22,25,30,32)(28,33,34,31)(35,36,40,44)(37,42,38,41)(39,47,48,43)(45,49,46,50);(4,8)(9,14)(10,15)(16,18)(22,30)(25,32)(28,34)(31,33)(35,37,40,38)(36,41,44,42)(39,46,48,45)(43,50,47,49);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34)(35,38,40,37)(36,42,44,41)(39,46,48,45)(43,50,47,49);(3,5,7,11)(4,9,8,14)(6,12,13,17)(10,16,15,18)(19,21,23,29)(20,24,26,27)(22,28,30,34)(25,31,32,33)(35,36)(37,42)(38,41)(39,43)(40,44)(45,50)(46,49)(47,48);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,22,23,30)(20,25,26,32)(21,28,29,34)(24,31,27,33)(35,39)(36,43)(37,45)(38,46)(40,48)(41,49)(42,50)(44,47)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008810);