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×#53364 |
| 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 | |
4098 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);(5,11)(9,14)(12,17)(16,18)(19,20,23,24)(21,25,27,31)(22,29,30,26)(28,34,33,32)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,50)(47,49);(4,8)(6,13)(9,14)(12,17)(20,24)(21,27)(22,30)(32,34)(35,37,40,38)(36,41,43,45)(39,44,47,46)(42,48,49,50);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,21)(20,25)(22,28)(23,27)(24,31)(26,32)(29,34)(30,33)(35,38,40,37)(36,41,43,45)(39,46,47,44)(42,48,49,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,20)(21,25)(22,26)(23,24)(27,31)(28,32)(29,30)(33,34)(35,39,40,47)(36,42,43,49)(37,44,38,46)(41,48,45,50);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,22,23,30)(20,26,24,29)(21,28,27,33)(25,32,31,34)(35,36)(37,45)(38,41)(39,42)(40,43)(44,50)(46,48)(47,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013115);