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×#54322 |
| 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 | |
4034 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);(3,4)(5,8)(6,11)(7,12)(9,13)(10,15)(14,16)(17,18)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,32,34,33)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,49)(48,50);(4,5)(6,13)(10,12)(14,18)(20,24)(22,31)(25,27)(28,34)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,5,8,4)(6,11,13,9)(7,12,15,10)(14,17,18,16)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,47)(48,50);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34)(35,37,39,45)(36,40,42,43)(38,44,47,50)(41,48,46,49);(3,7,8,15)(4,10,5,12)(6,14,13,18)(9,16,11,17)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)(35,38,39,47)(36,41,42,46)(37,44,45,50)(40,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007015);