Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53384 |
| 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)(21,27)(25,30)(28,33)(32,34)(35,36)(37,40)(38,43)(39,44)(41,45)(42,47)(46,48)(49,50);(4,8)(9,14)(10,15)(16,18)(20,24)(22,29)(25,30)(28,33)(36,37)(38,45)(42,44)(46,50);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34)(35,37,40,36)(38,43,45,41)(39,44,47,42)(46,49,50,48);(3,5,7,11)(4,9,8,14)(6,12,13,17)(10,16,15,18)(19,21)(20,25)(22,28)(23,27)(24,30)(26,32)(29,33)(31,34)(35,38)(36,41)(37,43)(39,46)(40,45)(42,48)(44,49)(47,50);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,22)(20,26)(21,28)(23,29)(24,31)(25,32)(27,33)(30,34)(35,39,40,47)(36,42,37,44)(38,46,45,50)(41,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013127);