Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61806 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6110 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);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,23,28)(21,30,32,25)(22,27,34,41)(24,26,36,29)(31,43,44,38)(33,39,46,37)(35,40,47,42)(45,49,50,48);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21,23,32)(20,25,28,30)(22,31,34,44)(24,33,36,46)(26,37,29,39)(27,38,41,43)(35,45,47,50)(40,48,42,49);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(20,26)(24,36)(25,37)(27,40)(28,29)(30,39)(33,46)(35,47)(38,48)(41,42)(43,49)(45,50);(5,13)(8,11)(12,18)(15,17)(19,22)(20,27)(21,31)(23,34)(24,35)(25,38)(26,40)(28,41)(29,42)(30,43)(32,44)(33,45)(36,47)(37,48)(39,49)(46,50);(4,10)(8,11)(9,16)(15,17)(20,28)(21,32)(22,34)(26,29)(33,46)(35,47)(38,43)(48,49)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485179);