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×#53363 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4132 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);(5,11)(9,14)(12,17)(16,18)(19,20,24,21)(22,27,29,25)(23,28,31,26)(30,32,34,33)(35,36,40,43)(37,38,45,41)(39,47,48,42)(44,49,50,46);(4,8)(6,13)(9,14)(12,17)(20,21)(22,29)(23,31)(32,33)(35,37,40,45)(36,41,43,38)(39,44,48,50)(42,49,47,46);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32)(35,37)(36,41)(38,43)(39,44)(40,45)(42,49)(46,47)(48,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34)(35,38,40,41)(36,37,43,45)(39,46,48,49)(42,44,47,50);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,23,24,31)(20,26,21,28)(22,30,29,34)(25,32,27,33)(35,39,40,48)(36,42,43,47)(37,44,45,50)(38,46,41,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013108);