Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3179 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5908 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)(19,20,23,27)(21,29,31,25)(22,33,35,49)(24,37,34,28)(26,40,41,36)(30,44,46,48)(32,39,45,43)(38,47,42,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,23,31)(20,25,27,29)(22,30,35,46)(24,32,34,45)(26,38,41,42)(28,39,37,43)(33,48,49,44)(36,47,40,50);(19,22,34,40,23,35,24,36)(20,26,37,49,27,41,28,33)(21,30,45,50,31,46,32,47)(25,38,43,44,29,42,39,48);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(21,31)(25,29)(30,46)(32,45)(38,42)(39,43)(44,48)(47,50);(4,10)(8,11)(9,18)(16,17)(20,27)(21,31)(22,35)(28,37)(32,45)(36,40)(38,42)(44,48)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485643);