Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3197 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20,24,28)(21,29,30,42)(22,26,33,40)(23,35,37,27)(25,36,38,32)(31,43,44,50)(34,47,46,41)(39,48,49,45);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(21,30)(25,38)(29,42)(31,44)(32,36)(39,49)(43,50)(45,48);(5,12)(8,11)(13,18)(15,17)(19,21,24,30)(20,25,28,38)(22,31,33,44)(23,36,37,32)(26,39,40,49)(27,42,35,29)(34,48,46,45)(41,50,47,43);(4,9)(8,11)(10,16)(15,17)(19,22)(20,26)(21,31)(23,34)(24,33)(25,39)(27,41)(28,40)(29,43)(30,44)(32,45)(35,47)(36,48)(37,46)(38,49)(42,50);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)(22,33)(26,40)(31,44)(34,46)(39,49)(41,47)(43,50)(45,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483048);