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×#54376 |
| 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 | |
4360 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)(13,17)(19,20,24,21)(22,25,29,27)(23,28,31,26)(30,33,34,32)(35,36,39,42)(37,43,44,40)(38,41,46,48)(45,49,50,47);(6,14)(10,15)(13,17)(16,18)(20,21)(22,29)(23,31)(32,33)(37,44)(40,43)(45,50)(47,49);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21,24,20)(22,25,29,27)(23,28,31,26)(30,32,34,33)(35,37)(36,40)(38,45)(39,44)(41,47)(42,43)(46,50)(48,49);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,17)(15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48)(49,50);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,23,24,31)(20,26,21,28)(22,30,29,34)(25,32,27,33)(35,38,39,46)(36,41,42,48)(37,45,44,50)(40,47,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008915);