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×#61814 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6164 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)(19,20,25,30)(21,26,34,44)(22,28,36,45)(23,29,24,39)(27,38,46,35)(31,47,32,41)(33,42,37,43)(40,48,50,49);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21,25,34)(20,26,30,44)(22,31,36,32)(23,37,24,33)(27,40,46,50)(28,47,45,41)(29,43,39,42)(35,49,38,48);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22)(20,27)(21,31)(23,35)(24,38)(25,36)(26,40)(28,39)(29,45)(30,46)(32,34)(33,48)(37,49)(41,43)(42,47)(44,50);(5,13)(8,11)(12,18)(15,17)(19,22)(20,28)(21,32)(23,38)(24,35)(25,36)(26,41)(27,39)(29,46)(30,45)(31,34)(33,48)(37,49)(40,43)(42,50)(44,47);(4,10)(8,11)(9,16)(15,17)(19,23,25,24)(20,29,30,39)(21,33,34,37)(22,35,36,38)(26,42,44,43)(27,45,46,28)(31,48,32,49)(40,47,50,41)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485189);